Let $X$ be a "nice" space: metrizable, connected, locally path connected perhaps. Let $K\subset X$ be a compact set.
Is there a always a compact connected $L\subset X$ such that $K\subset L$?
This is true if we assume local compactness: cover $K$ with a finite number of connected relatively compact open sets, take their closure, and then join with arcs. However, without local compactness I don't know what to do.
Choose a sequence $\varepsilon_n\to 0$ and a $\varepsilon_n$-net $N_n$ for each $n$. Assume $N_0$ is a one-point set. For each point in $x\in N_k$ choose a closest point in $y\in N_{k-1}$ and connect $x$ to $y$ by a curve. Note that we can assume that diameter of the curve is at most $\delta_k$ for a fixed sequence $\delta_k\to 0$.
Consider the union $K'$ of all these curves with $K$; observe that $K'$ is compact and path connected.
This is meant to fill in some of the details outlined by Anton Petrunin's answer, and also to refine the statement slightly. Recall that a compact connected Hausdorff space is called a continuum.
We will call a topological space $X$ continuum-wise connected if every $x,y\in X$ can be joined by a continuum, i.e. there is a continuum $K\subset X$ that contains both $x$ and $y$. We will call $X$ locally continuum-wise connected if for every $x\in X$ and open neighborhood $U$ of $x$ there is an open neighborhood $V$ of $x$ such that every $y\in V$ can be joined by a continuum within $U$. It is easy to see that continuum-components of locally continuum-connected space are open and disjoint, and so a connected locally continuum-connected is continuum-connected.
Proposition. A metrizable space $X$ is locally continuum-wise connected if and only if there is a metric $\rho$ on $X$ compatible with the topology and such that every open ball of radius less than $1$ is continuum-connected.
This is analogous to Theorem IV.7.1 in Newman - Elements of the topology of plane sets of points. There it is stated for (locally) connected metrizable spaces, but works also for any (locally) set-wise connected metrizable spaces, for an appropriate collection of connected sets (e.g. separable, bounded, arcs).
Proof. Sufficiency is clear. Let us prove necessity. Choose an arbitrary metric $d$ on $X$ bounded by $1$. For $x,y\in X$ declare $\rho(x,y)$ to be the infimum of diameters of the continuums that join $x$ and $y$ (if $x$ and $y$ are not joined by any continuum put $\rho(x,y)=1$). It is easy to see that $\rho$ is a metric, and moreover $d\le\rho$. Furthermore, if $x_n\to x$, since $X$ is locally continuum-wise connected, $x_n$ and $x$ can be joined by arbitrarily small continuums, and so $\rho(x_n,x)\to x$. Thus, $\rho$ is equivalent to $d$, and so is compatible with the topology of $X$.
It is left to show that every ball of radius less than $1$ is continuum-wise connected. Let $x\in X$ and let $R<1$. Assume that $y\in B_{\rho}(x,R)$, i.e. $\rho(x,y)=r<R<1$. By definition of $\rho$, there is a continuum $K$ with $d$-diameter at most $\frac{r+R}{2}$ that joins $x$ and $y$. Every point $z\in K$ is joined with $x$ by $K$, and so $\rho(x,z)=\frac{r+R}{2}<R$. Hence, $K\subset B_{\rho}(x,R)$, and so $y$ is joined by $x$ by a continuum in $B_{\rho}(x,R)$. $\square$
Corollary. A metrizable space $X$ is locally continuum-wise connected if and only if every point has a base of open continuum-wise connected neighborhoods.
Now, having these characterizations we can answer the original question.
Theorem. Let $X$ be a connected and locally continuum-wise connected metrizable space. Then for every compact $K\subset X$ there is a continuum $L\subset X$ that contains $K$.
Before proving the theorem, let us prove the following characterization of compactness.
Lemma Let $Y$ be a metric space for which there is a compact $K\subset Y$ such that for every $\varepsilon>0$ there is a compact $N$ such that $K$ is an $\varepsilon$-net of $Y\backslash N$. Then $Y$ is compact.
Proof. It is clear that $Y$ is completely bounded. We only need to prove completeness. Let $\{y_m\}\subset Y$ be a Cauchy sequence. It is enough to find a convergence subsequence. For every $k$ let $N_k$ be compact and such that $K$ is $\frac{1}{k}$-net for $Y\backslash N_k$. We may assume that $N_k\subset N_{k+1}$.
If an infinite subsequence of $\{y_m\}$ was contained in $N_k$, for some $k$, then there would be a convergent subsequence due to compactness of $N_k$. Hence, we can choose a subsequence $\{z_m\}$ such that $z_m\not\in N_m$. Since $K$ is an $\frac{1}{m}$-net for $Y\backslash N_m$, there is $x_m\in K$ with $\rho(x_m,z_m)<\frac{1}{m}$. Since there is a subsequence of $\{x_{m_k}\}$ that converge to $x\in K$, so does $\{z_{m_k}\}$. $\square$
Proof of the theorem. Using the proposition, we can metrize $X$ with a metric such that open balls of radius less than $1$ are continuum-wise connected.
For natural $n$, let $K_n\subset K$ be a finite $\frac{1}{2^n}$-net of $K$. For every $x\in K_{n+1}$ there is $y\in K_{n}$ such that $\rho(x,y)<\frac{1}{2^n}$. Since $B(y, \frac{1}{2^n})$ is continuum-wise connected, there is a continuum $L^n_{x}\subset B(y, \frac{1}{2^n})$. Then for any $m>n$ and $x\in K_m$ and $z\in L_x$ there $y\in K_{n}$ such that $\rho(z,y)<\frac{1}{2^{n-1}}$.
Let $z\in K$ and for $x\in K_1$ let $L^1_x$ be a continuum that joins $x$ with $z$. Observe by induction that $M_n=\bigcup_{i\le n, x\in K_n} L_{x}^i$ is a continuum, and so $M= \bigcup M_k$ is connected. Since $M$ contains an $\frac{1}{2^n}$-net of $K$, for every $n$, it follows that $K\subset \overline{M}$. Hence, $M\subset M\cup K\subset \overline{M}$ from where $Y=M\cup K$ is connected.
Finally, since $K_n\subset K$ is a $\frac{1}{2^{n-1}}$-net for $K\cup \bigcup_{k>n} M_k\supset Y\backslash M_n$, for every $n$, $Y$ is compact due to the Lemma.$\square$
Remark. I also would like to present a nice example that bof gave in the comments (now deleted), that at least local connectedness is required: Consider the following modification of the topologist's sine curve $X=\{(t,\sin \frac{1}{t}), 0<t\le 1\}\cup\{(0,0\}$, which is connected and moreover is a polish space. However the compact set $\{(x,y)\in X, y=0\}$ cannot be connected by a continuum. Note that for a completely metrizable space local connectedness is equivalent to local path-connectedness.
Here is another answer, based again on Anton Petrunin's idea, but obtaining a slightly different result.
Theorem. Let $X$ be a connected and locally path-connected completely metrizable space. Then for every compact $K\subset X$ there is a Peano continuum $L\subset X$ that contains $K$.
In order to prove this result we need a version of the proposition from my previous answer.
Proposition. A locally path-connected completely metrizable space $X$ supports a complete metric $\rho$ on $X$ compatible with the topology and such that every open ball of radius less than $1$ is path-connected.
Proof. Let $d$ be a complete metric on $X$ bounded by $1$. Apply the same construction as in my previous answer (but with paths instead of continuums) and obtain $\rho$. Since $\rho\ge d$ are equivalent, and the latter is complete it is easy to see that the former is also complete (a $\rho$-Cauchy sequence is a $d$-Cauchy sequence, hence it is $d$-convergence, and so $\rho$-convergent).$\square$
Proof of the Theorem. We will construct a convergent sequence of paths $\varphi_n:[0,1]\to X$ such that the image of $\gamma_n$ contains a $\frac{1}{2^n}$-net of $K$.
For natural $n$, let $K_n\subset X$ be a finite $\frac{1}{2^n}$-net of $K$. Using connectedness one can choose them so that $K_n\cap K_m=\varnothing$. Moreover, let $K_1=\{x_0,...,x_n\}$.
Let $\gamma_1:[0,1]\to X$ be a continuous path such that $\gamma_1|_{[\frac{2i}{2n+1},\frac{2i+1}{2n+1}]}\equiv x_i$, $i=0,...,n$ (on the intermediate segments $\gamma_1$ joins $x_i$ with $x_{i+1}$, which is possible since $X$ is path connected).
For $0\le a<b\le 1$ and $x,y\in X$ with $\rho(x,y)<r<1$ let $\gamma:[a,b]\to X$ be a a continuous loop such that $\gamma(a)=\gamma(b)=x$, $\gamma|_{[\frac{2a+b}{3},\frac{a+2b}{3}]}\equiv y$, and the image of $\gamma$ is contained in $B(x,r)$ (which is possible since open balls of radius less than $1$ are path connected).
Now assume that $\gamma_n$ is constructed so that its image contains $K_n$ and for every $x\in K_n$ there are $a<b$ such that $[c,d]\subset \gamma^{-1}_n(x)$. Let $x_1,...,x_m\in K_{n+1}$ be such that $\rho(x_i,x)<\frac{1}{2^n}$. Re-define $\gamma_n$ on $[c,d]$ to be a series of loops defined above from $x$ to $x_1$ and back, then from $x$ to $x_2$ and back, and so on.
Applying the same construction to all elements of $K_n$ (simultaneously) we get $\gamma_{n+1}$ such that for every $y\in K_{n+1}$ there are $a<b$ such that $[a,b]\subset \gamma^{-1}_{n+1}(y)$. Moreover, if $\gamma_{n+1}(t)\ne \gamma_n(t)$, it follows that $\gamma_{n+1}(t)\in B(\gamma_n(t),\frac{1}{2^n})$, from where $\rho(\gamma_{n+1},\gamma_n)\le \frac{1}{2^n}$.
Note that the image of $\gamma_{n+1}$ contains the image of $\gamma_n$. Moreover, from construction and the fact that $K_m$'s are disjoint that if $x\in K_n$, then $\gamma_{n+1}^{-1}(x)\ne \varnothing$ and if $t\in \gamma_{n+1}^{-1}(x)$, then $\gamma_m(t)=x$, for all $m>n$.
It follows that $\{\gamma_n\}$ is a Cauchy sequence of maps from $[0,1]$ into a complete space $X$. Consequently, it uniformly converges to a map $\gamma:[0,1]\to X$. From the previous paragraph it follows that the image of $\gamma$ contains every $K_n$, and since it is compact, it contains $\overline{\bigcup K_n}\supset K$. $\square$
