We have been in the beta for more than 700+ days. What is the procedure and requirements to get out of the Beta stage?
There is no clear rule for this. Some sites have remained in beta for several years. For example, Board & Card Games has been in beta for 8 years and 7 months, since October 2010.
Being in beta means we don't have the same reputation thresholds for the various site privilege as graduated sites. Which is a very good thing, we aren't ready for that yet. We only have one user at >10k rep (which is what gives access to the moderator tools on graduated sites, it's 2000 for betas), no users past 20k rep (the threshold for full access, 5k on betas). Most importantly, we only have 9 users with >3000 rep, which is what is needed to be able to cast close and reopen votes, the most essential part of site maintenance.
So, if we were to graduate tomorrow, there simply wouldn't be enough users with the required rep to do site maintenance. But this isn't a problem! It doesn't mean we are at risk of losing the site, we can happily stay in beta for as long as needed.
Some relevant discussions on the main meta:
Just to note, the number of questions per day is steadily rising, or perhaps more accurately has recently jumped. When I first joined the site, not so long ago (less than 6 months), it was 2.2/2.4 questions per day and has currently jumped to 2.8 questions per day. As long as that trajectory continues, then this is a stable beta, which in the fullness of time would be expected to be a full site. I don't think there is any reason to hurry.
As of today it is 3.3 questions per day ... its a non-linear function, my favourite
Woooaaa wait here today September 26, 2019
The number of vists is 1,517 (!!) ... so the site now has two scores of "Excellent".
The number of questions is 3.1 per day ... so whilst ever it on the trajectory towards a full launch its is here to stay.