In TOS, shields provide very good protection for ships and planets against weapons fire of all sorts. They tend to be curiously transparent to anything termed radiation, however. It is difficult to gauge what is required to bring down a shield; the looseness of estimates of weapons firepower lead to greater looseness in estimates of shielding.
In "Whom Gods Destroy," we see a planetary shield that can resist a narrow beam phaser attempting to punch a hole in it. If two narrow beam phaser blasts at full power have enough energy to punch a hole its own diameter in the shield, and the rest of the shield would remain intact, then we could generously estimate the shield's total energy capacity at 4x1033 to 8x1034 joules.
A more conservative measurement of the same incidentwould be to examine the intensity reduction of weapons striking the shield. A narrow beam phaser strike should be no wider than the beam seen in "The Alternative Factor," or roughly a tenth meter across. With a 1-10 gigaton blast* falling in a 0.008m2 circle over somewhere between a half and a tenth second, this gives us a blocked intensity of ~1-50 zettawatts per square meter.
In "Balance of Terror," a plasma torpedo with a yield estimated at ~100 gigatons* appeared to threaten instant death to the Enterprise. We could guess that to be an outer limit of its shields accordingly. On the lower end, a single phaser blast potentially threatens instant death to a warship at 1-10 gigatons.
In "Changeling," we discover that the Enterprise's shields may weather five energy bolts with 90 times the energy of a photon torpedo, each dropping the shields by 20%. We have no independent estimates on what yield photon torpedos are in TOS itself, but if we consider a phaser blast an acceptable substitute for a photon torpedo (based on the reference to photon torpedos as a possible brute force solution in "Whom Gods Destroy") then this would place shield capacity at 450-4500 gigatons. If we place photon torpedos as on par with an impulse engine detonation, this would give us a shield capacity of ~45 gigatons.