The theorem that inkspot refers to in their answer is originally from Inoue's paper New Surfaces with No Meromorphic Functions, II which seems like a more complete reference for this question. In particular, Inoue gives an explicit example of a compact complex surface which has an analytic subvariety but no compact complex submanifolds.
If $x$ is a real quadratic irrationality (i.e. a real irrational solution of a real quadratic equation), denote it's conjugate by $x'$. Let $M(x)$ be the free $\mathbb{Z}$-module generated by $1$ and $x$, then set $U(x) = \{\alpha \in \mathbb{Q}(x) \mid \alpha > 0, \alpha\cdot M(x) = M(x)\}$ and $U^+(x) = \{\alpha \in U(x) \mid \alpha\cdot\alpha' > 0\}$. Both $U(x)$ and $U^+(x)$ are infinite cyclic groups and $[U(x) : U^+(x)] = 1$ or $2$.
If $\omega$ is a real quadratic irrationality such that $\omega > 1 > \omega' > 0$, then $\omega$ is a purely periodic modified continued fraction; that is, $\omega = [[\overline{n_0, n_1, \dots, n_{r-1}}]]$ where $n_i \geq 2$ for all $i$, $n_j \geq 3$ for at least one $j$, and $r$ is the smallest period. For every such $\omega$, Inoue constructs a compact complex surface $S_{\omega}$ which is now known as an Inoue-Hirzebruch surface.
There are compact subvarieties $C$ and $D$ of $S_{\omega}$ with irreducible components $C_0, \dots, C_{r-1}$ and $D_0, \dots, D_{s-1}$ respectively; here $s$ is the smallest period of the modified continued fraction expansion of another element $\omega^*$ related to $\omega$ (alternatively, $s$ can be determined from the modified continued fraction expansion of $\frac{1}{\omega}$). When $r \geq 2$, $C$ is a cycle of non-singular rational curves, and when $r = 1$, $C$ is a rational curve with one ordinary double point. Proposition $5.4$ shows that $C_0, \dots, C_{r-1}, D_0, \dots, D_{s-1}$ are the only irreducible curves in $S_{\omega}$.
In the case where $[U(\omega) : U^+(\omega)] = 2$, we have $r = s$. Furthermore, there is an involution $\iota$ such that $\iota(C_i) = D_i$ for $i = 0, \dots, r - 1$. The quotient of $S_{\omega}$ by $\iota$ is denoted $\hat{S}_{\omega}$ and is now known as a half Inoue surface. Note that the images of $C_0, \dots, C_{r-1}$ are the only irreducible curves in $\hat{S}_{\omega}$.
If we can find a real quadratic irrationality $\omega$ such that $\omega > 1 > \omega' > 0$, $r = 1$, and $[U(\omega) : U^+(\omega)] = 2$, then $\hat{S}_{\omega}$ is a compact complex surface containing a unique curve, namely a rational curve with one ordinary double point. In particular, it provides an example of a compact complex manifold with a subvariety but no compact complex submanifolds. One such $\omega$ was given in the paper (end of section 6).
Example. Take $\omega = (3 + \sqrt{5})/2$. Then $[U(\omega) : U^+(\omega)] = 2$ and $\alpha_0 =\ \text{a generator of}\ U(\omega) = (1 + \sqrt{5})/2$, $\alpha = \alpha_0^2 = (3 + \sqrt{5})/2$, $\omega = [[\overline{3}]]$, $r = 1$.
In this case, $b_2(\hat{S}_{\omega}) = 1$ and $\hat{S}_{\omega}$ contains exactly one curve $\hat{C}$. Moreover, $\hat{C}$ is a rational curve with one ordinary double point and $(\hat{C})^2 = -1$.
For those interested in the details, in addition to Inoue's paper, it may also be worth reading the earlier paper Hilbert modular surfaces by Hirzebruch. As mentioned in his paper, Inoue used some methods from Hirzebruch's paper (which gives some indication of why the resulting surfaces are jointly named).
