Will I be able to use Blender 2.8 and/or Eevee?

My graphics card is not supported currently for GPU rendering (Blender 2.79a). Does this mean that I won’t be able to use Blender 2.8 when it comes out? What about using Eevee rendering? I use Blender only for my own enjoyment, so I don’t think I’d want to spend several hundred dollars for a new GPU.

Cycles render requirements are not necessarily related to OpenGL requirements for Eevee in 2.8. What graphic card do you have?

AMD ATI Radeon HD 5570

You can use 2.8 and Eevee with that card. Maybe a little slow in Eevee, but meets OpenGL requirements.
Have you tried Blender 2.8 at the moment? You even have sample scenes here:

Be careful not to open and save with the same name in 2.8 your previous works made with stable versions of Blender.

Thank you. I will try.