We will call a topological space $X$ continuum-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-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-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-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-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-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-connected if and only if every point has a base of open continuum-connected neighborhoods.

Theorem. Let $X$ be a connected and locally continuum-connected metrizable space. Then for every compact $K\subset X$ there is a continuum $L\subset X$ that contains $K$.

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-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-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 $\rho(z,y)<\frac{1}{2^{n-1}}$.

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.

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.

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 $K$.

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 $\rho(z,y)<\frac{1}{2^{n-1}}$.

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.

This is meant to fill in some of the details outlined by the Anton Petrunin's answer, and also to refine the statement slightly. Recall that a compact connected Hausdorff space is called a continuum.

If an infinite subsequence of $\{y_m\}$ was contained in $N_k$, for some $k$, then there would be 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 opens balls of radius less than $1$ are continuum-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.

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-connected.