Examples of Manifolds such that $\chi (X)=-3$

Some points.

1) An oriented manifold $M$ which occurs as the boundary of an oriented manifold $W$ necessarily has $\chi(M)$ even. This means that not only are all of your surfaces not going to work, but neither is anything that's the boundary of a manifold.

2) Every odd-dimensional manifold has $\chi(M) = 0$ (even non-oriented ones). So you can't try that.

3) $\chi(M \# N) = \chi(M) + \chi(N) - 1 - (-1)^n$ where $n$ is the dimension of $M$ and $N$. (Since we'd better be working with even-dimensional manifolds, this just says $\chi(M \# N) = \chi(M) + \chi(N) - 2$.) This says, then, if you can find a single manifold with euler characteristic 1, you can find one with Euler characteristic -3, either by connect-summing it with itself four times or by connect-summing something of euler characteristic zero twice.

So you just need to find a nice manifold with Euler characteristic 1. (Actually, any positive odd number will do.) The first place you can possibly do this is in dimension 4. Do you know any nice manifolds there?

By the classification of surfaces, every closed oriented surface is a connected sum of tori, so you won't be able to find a $2$-dimensional example. By Poincare duality, any closed odd-dimensional manifold has Euler characteristic $0$, so the smallest dimension where you can find an example is $4$.

To find a $4$-dimensional example, you can start with $\mathbb{C}P^2$, which has Euler characteristic $3$. Can you see a way you can modify a closed oriented $4$-manifold to decrease its Euler characteristic by $2$? Then you just have to repeat this three times to get a $4$-manifold of Euler characteristic $-3$.