Lesson 54: "Oh well" and "I can't help it" with "It can't be helped"
There's two basic forms of this expression, differing in whether they use よう・様 or かた・方.
しようがない・しょうがない・etc
しかたがない・しかたない・etc
Both of them are literally し(する) + よう・かた (way to X) + [が]ない.
The literal meaning of ～よう・かたがない is "there's no way to X". しよう・かたがない can use this meaning, too, in the right context.
書きようがない No way to write
But as a set phrase, しようがない and friends are their own thing, and take on the meaning "oh well", "I can't help it", "there's no two ways about it", etc. English has a bunch of phrases for this kind of idea with different nuances.
This sort of expression shows up in every human language, but the way it's worded in japanese has been exoticized, thanks to shitty translations repeatedly using "it can't be helped".
While grammatically correct, and a valid phrase, and one that people might use in natural english, overusing "it can't be helped" in translations has basically made しようがない have a mythical status as something that has an untranslatable meaning. It's not untranslatable. Especially because many uses of しようがない just mean "there's no way to do X". Not all of them are "oh well".
There's something very important to learn here. Your dictionary is not an authority on what something does mean, only an authority on what something might mean. Just because しようがない has the translation "it can't be helped" in a dictionary, it doesn't mean that you must interpret it that way. What you feel like it means is more important than what your dictionary says.