Depending on what you mean, any language can.
For example, C++ can. At first sight, this is absurd - C++ is a statically typed compiled language. So - what you do is include the LLVM library in your project. This is a compiler back-end, and you can use this to describe your classes, compile them, and run them using the LLVM JIT, all at run-time for your application.
IIRC, the gcc back end is written in C, so if you're willing to figure out that code, you could in principle define classes at run-time using a language that doesn't even have classes.
Either way, part of your job is to define what exactly a class is - that isn't built into the compiler back ends as they are supposed to support a range of different front-end languages with different type systems.
Of course I'm not recommending this approach - just pointing out that it's possible.