There is a big difference betweent the 1903 Springfield and the 1903-A3 Springfield. The first was manufactured and used in WWI, and thereafter, and the second was not manufactured and used unti the start of WWII. The seconded has a great peepsight, but was more cheaply made, using stamped parts, and used very hard steel, so the bolt (opening and closing) always drags and feels "sticky" (because it doesn't "wear in" and there is not much yolu can do about it). However, they are very strong rifles.
The older rifle, being the original 1903, is made of a slightly softer metal, and contains all milled parts. It is considered the better of the two rifles, BUT (i) has very small tangent sights like the 98 Mauser, which are really hard to line up except in a clear open field, and (ii) during WWI, rookie metal-smiths were added to the workforce at the Springfield and Rock Island arsenals, and proceeded to heat the receivers to way too high of a temperature, ruining the temper and making the actions brittle. How many thousands did they ruin this way? Nobody knows. Reports started coming in from the trenches of exploding actions, and actions breaking when struck with hard objects.
The military found out what was causing the problem and proceded to correct the procedures at the arsenals. However, to be sure that you do not get an action that is potentially defective (read that as dangerous), do not buy an 03 Springfield made at the Springfield arsenal unless it has a serial number above 800,000 and do not buy one made at the Rock Island arsenal unless it has a serial number above 400,000.