Can we solve the three-body problem with computers?

There are no exact general solutions, with or without computers. There are a few special cases with solutions. In practical physical systems, we have simulations which are accurate to 14 or more decimal places, which is more than good enough.

We can simulate in arbitrarily small time intervals but that is not the same as a general case mathematical solution where each body's location is given mathematically as a function only of time.

Do you mean "free-body problem"?