This phrase has very much the same kind of feeling as brace yourself, albeit perhaps slightly more literary, and not with the same sense of immediacy that brace necessarily infers. Steel yourself is a warning to prepare your spirit for some kind of hardship to come. (It can, of course, also be used ironically when the difficulty is only slight.)
Collins gives the following definition:
to prepare (oneself) for coping with something unpleasant
I was steeling myself to call round when Simon arrived.
Synonyms of steel yourself
He braced himself for the icy plunge into the black water.
grit your teeth
Oxford Living Dictionaries defines steel as follows:
verb [with object]
Mentally prepare (oneself) to do or face something difficult.
‘his team were steeling themselves for disappointment’
with infinitive ‘she steeled herself to remain calm’
The sense is, of course, to harden yourself and make yourself like steel. The Online Dictionary of Etymology attests to the verb steel being used to mean make hard or strong like steel in the 1580s. (It could be used with a fair amount of thematic consistency along with the word mettle, if you're that way minded.)
Shakespeare used the verb steel in this sense on more than one occasion.
In Henry VI Part II, he gives the following lines to Richard Plantagenet, Duke of York, in a pep-talk soliloquy. Plantagenet (or York) is plucking up his courage for the task at hand - putting down a revolt in Ireland in the name of the King and, then, usurpation of the English throne for himself:
Now, York, or never, steel thy fearful thoughts,
And change misdoubt to resolution:
Be that thou hopest to be, or what thou art
Resign to death; it is not worth the enjoying:
Let pale-faced fear keep with the mean-born man,
And find no harbour in a royal heart.1