Steel stair stringer size
Yes, two 5” x 2” x 1/4” steel tube stringers  (with A36 steel) spanning 10’ will adequately support your stairway.
Code requires a Live Load of between 40 lbs. per square foot (psf) and 100 psf depending where it’s located. Assuming it’s 100psf x 3’ wide x 10 long = 3,000 lbs. plus a Dead Load of 15 psf x 3’ x 10’ = 450 lbs. for a Total load of 3,450 lbs.
Your tubes will support about 8,000 lbs. each, depending on connections, etc.
If you don’t have the tubes already, you may want to consider a 5” x 1 3/4” channel which will support 5,100 lbs. each.