Yes. I know people who have taken cross country trips on 50cc scooters.
It's best to stay off the highways though. Some people would feel more secure on a 500. Both the 500 and 250 have relatively upright riding positions, but may be tiring on long hauls. If I may make a suggestion?
People have gone round the world on BMW f650gs motorcycles. They have Upright seating position, ability to carry mucho luggage and can tackle most any kind of terrain. They also have a plethora of aftermarket for touring - from fitted luggage to more comfortable seats, and can be fitted w/ extra large fuel tanks for the long haul. BMW is also well known for their excellent customer service and support around the globe.
Problem is, they may be out of your price range w/ an MSRP of $8,890 for the base model.
This brings me to a secondary option - The Kawasaki KLR650. Same type of bike, same size engine, same amount of aftermarket support and same long distance capabilities. I would dare say more people have taken these around the globe than the BMW - It's been around longer and it's cheaper. With an MSRP of $5100 it's in the range of the ex500, but it's got better long distance capabilities (Both in passenger comfort and on/off road capabilities.)
Something to consider anyway

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