Yes, there are lots of espresso machines for home use. I have two suggested routes for you to choose:
Home Espresso route
Personally I've used high-end machines like Vibiemme Domobar Junior to low-end ones like the Solis.
Now I will say that I roast and grind my beans at home so it's pretty much like a home cafe (with the easy bowel movements!) - you don't have to do this you can just buy filter coffee from the store or online then it will depend on Noorjahan's first point about coffee bean selection...
Alternative Route
You could instead make turkish coffee on the stove, there are lots of videos on youtube of how to do it, like the one titled "Coffee Tips, How To Prepare Turkish Coffee Using A Cezve?" Note instead of using the turkish pot in those videos, I use a small 750ml to 1ltr saucepan, it works just as well. I also sieve the bits out with a very fine filter.
Just make sure no one is using the bathroom when you have this! You will immediately need to go!
You could try this second route and then if the taste is not to your liking you could try the home espresso route.
On another note
You said you have a French press. I feel it should still have enough kick to get the bowels moving, anyone else agree or disagree? Perhaps you could try different beans and let us know if this works.
Hope the above helps.