The continuous image of a compact subset of a topological space is itself compact. Try and find some compact set S in R^n for some n, and some map f:R^n -> M(3,C) where C is the complex numbers. If you restrict this map to S, the map should have SU(3) as its image. From this, it follows that SU(3) is a compact subset of M(3,C).
Hint, it might be easier to think of R^n as being C^m for some m, seeing as the variables in the elements of SU(3) are complex, you can then just associate your map f(x) as f(g(x)) where g is your standard homeomorphism from R^2m to C^m.