It's necessary to read the full description:
1 Kings 7:23 And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.
7:24 And under the brim of it round about there were knops compassing it, ten in a cubit, compassing the sea round about: the knops were cast in two rows, when it was cast.
7:25 It stood upon twelve oxen, three looking toward the north, and three looking toward the west, and three looking toward the south, and three looking toward the east: and the sea was set above upon them, and all their hinder parts were inward.
7:26 And it was an hand breadth thick, and the brim thereof was wrought like the brim of a cup, with flowers of lilies: it contained two thousand baths.
It helps to understand that the sea has a thickness of a handbreadth, and that we can use this to determine the ratio between a cubit and a handbreadth used.
There's a circle with circumference 30 cubits on the inside, and a circle with diameter 10 cubits around the brim.
Let's call the radius of the inner circle, r, and the outer circle R, and let's use h for the handbreadth, all in cubits.
and substituting in second equation 2π(R-h)=30
To rearrange in terms of h, first divide by 2π, so R-h=30/2π
then add h-30/2π, so R-30/2π=h
and substituting in formula for h gives: h=5-30/2π
and simplifying, h=5-15/π=0.225351707243... cubits
Which gives us about 1/h=4.43750798356... handbreadths in a cubit.
Now allegedly cubit comes from a word meaning elbow, and cubit bone refers to what we now call the ulna, a bone in the forearm. A cubit of 4.43 handbreadths would correspond to a close fisted cubit, meaning a measure from the elbow to the knuckles. (Side note: a cubit arm in heraldry is usually close fisted.)
One can verifiy this is approximately correct by counting how many handbreadths there are from one's elbow to one's knuckles. It should be about or just under four and a half. To measure more accurately one would need to take measurements from many people to get an average
So there does not appear to be any large imprecision in the measurements, and π≠3.
Now, let's ask how many fingers in a cubit.
Defining a finger to be a quarter of a cubit gives us:
4/h=17.7500319342... fingers in a cubit
Now that is very close to 17.75=17¾=71/4, so let's assume that is, or is an approximation to, how the cubit is defined: 71/4 fingers or 71/16 handbreadths i.e. h=16/71. (Remember that the sea is 10 cubits across so an error of 1/4 fingers becomes 10/4 fingers or 10π/4 fingers (nearly two handbreadths) in circumference. Using 18 fingers in a cubit would be too imprecise.)
Working backwards to give us an approximation for π we start from:
2π(R-h)≈30 and h=16/71
so π≈355/113=3.14159292035... (cf π=3.14159265359)
which is accurate to 7 significant figures or less than one part in ten million.