I wouldn't do it now. Neither would most people. The 1903A3 rifles have become collector's items. But in 50 years ago they were common and a lot of old rifles were taken and modified. Here's some articles from the August and September 1957 issues of The American Rifleman on how to make an "economy sporter, deluxe sporter, or match rifle". How to cut down the stock, re-barrel the the action, drill and tap the receiver and more.
You still see these from time in gun shops or on-line auctions. Auction Arms has one currently for sale. The Buy It Now price is $425.00. It might be a fine shooter, but that's less than half what it would be worth if it had just been left unmodified.
Here's a picture from the auction. Bonus points for picking out what's wrong in the picture.