I think that the normal US Open field, counting the qualifying folks is over 7,000. You have to have a handicap index of 1.4 as mentioned above but that only gets you into the first level of qualifying. If you move though that, generally about 10% make it though that level, then you move into sectional qualifying along with all the folks who made it plus those exempt from the first level, tour pros, etc. I think that around 5% of those folks make it into the open. Generally about 50% of the 156 person field had to go through either one or two rounds of qualifying.
Years ago I had the handicap index and tried to qualify. I think it was 2.8 then and played with a friend who was a plus 2 index. The qualifying courses are set up very difficult and very long. I manged to shoot 83 and 81 in my two rounds, which put me way out of any chance to qualify. I think the 3 people from our site shot around 6 under for the two rounds. My attempt got me a letter from the USGA in which I was asked to explain who I shot 12 over par for 2 rounds with an index of 2.8, standard from them when you exceed some magic number. My response was that I played badly lol They do this to keep folks from going out to play on a good course with a practice round for the sum of $50, at the time. Hey I thought 3 rounds at Pinehurst for $50 was fair.