I make espresso with whatever coffee I have on hand all the time. Whether that is proper or ideal I don't know but I like drinking it.
I've tried a number of inexpensive espresso machines & gave up. This is definitely a case of you get what you pay for.
I give this post a big +1. The grinder is much more important than the espresso machine when making espresso. You can get good results with a used Gaggia (carezza/coffee/baby) and a used burr grinder (nemox lux/Gaggia MDF). No matter what anyone tells you, an Aeropress will make a mean cup of coffee, but it will never make a true espresso. Good luck!Have you read the Espresso articles on Sweet Maria's? They confirm that buying a best grinder you can afford is critical, much more important than the espresso machine. I own a Rocky grinder (outstanding) and recently bought a refurb espresso machine from Whole Latte Love. But, these are well above your limit. Cheap shots of espresso are a waste of money. You might be better off buying an AeroPress and just drinking the initial press without diluting it. This is NOT espresso, but you'll get some flavor, and be well under your budget.
Mike