If I want to copy/paste a picture from some web site, and the picture is 250x250 pixels, is there some method to enlarge it to 1000x1000 pixels without losing sharpness? Here's an example of a 250x250 pixel picture that needs enlargement.

There is no such method that I'm aware of. "Zoom and Enhance"

You could contact the original photographer and ask for a larger image when you request permission to use their work.
That has worked for me in the past. Most people love to share as long as their name is spelled correctly.

The name I had was C. Peter's Creek. This picture originated at "OrchidWiz" thanks to Jim, so I don't know how I or Jim could actually contact the photographer. So far, with this particular orchid, "OrchidWiz" is the only known source.

UPDATE:

I just subscribed to AOS Orchids Plus and found the orchid to be C. Peter's Creek 'Ponkan' AM/AOS. There is a better picture (more pixels) of that orchid there and I have attached it here, as well as the one sourced from OrchidWiz.

AOS Orchids Plus

OrchidWiz

By clicking on the pictures, you can see the difference that increasing the pixels does.

Another way is to print the picture with highest quality then scan it at a very high resolution and higher percentage rate.

Thanks, Anne.

You're welcome! That's an ancient film photographers trick - every time I smell vinegar it reminds me of the olden days when you spent more time in the darkroom than out shooting pictures.

You beat me to it, Anne -- I've done that a lot of times with small photos that people have sent me!
(That, and a little Photoshopping, can do wonders!! )