The phrase not to be able to fight one’s way out of a paper bag, and variants, mean:
– to be extremely weak or incompetent at fighting;
– to be completely ineffectual or inept.

All the earliest occurrences of the phrase that I have found are Australian English.
—In chronological order:

1-: From Items of Interest, published in The Armidale Chronicle (Armidale, New South Wales) of Wednesday 8^th January 1902:

A little man, inflamed by wine and the geniality of the season of peace on earth and goodwill to men, wanted to fight a good-natured giant. “Fight, you little insect?” said the man of inches, “why, you could not fight your way out of a paper bag.”

2-: From The Sun (Kalgoorlie, Western Australia) of Sunday 27^th December 1903:

At Anaconda.
Things Only Middling.
With the Miners at Murrin.

“Miner” writes:—I have been waiting to see whether anyone would draw attention to the chaotic state of the Murrin Copper Mines Ltd, Anaconda. […]
I was fortunate in obtaining underground work the first time of asking, starting on afternoon shift. […]
[…]
On my shift were five or six men of the same name as the shift boss. Found out they were either brothers or sons or uncles of the said shift boss. One of them I recognised as a beer puller in one of the hotels, but got on as a miner at 13s. 4d, another lad who seemed not able to fight his way out of a paper bag was also working at 13s. 4d.—also a son.

3-: From the Narrogin Advocate and Southern Districts Courier (Narrogin, Western Australia) of Wednesday 14^th September 1904—the noun beer-sparrer designates a heavy drinker of beer:

Answers to Correspondents.

“Anti-Smooge”—Yes, you are about right, but we really have neither inclination nor space to publish your open letter, as the individual you refer to in such complimentary (sic!) terms is not of sufficient importance (in our opinion, anyhow) to waste time over. We have no time for beer-sparrers or toadies. The worm in question has not sufficient brains to give him a headache, nor could he fight his way out of a paper bag, whilst he has a sufficient number of aliases for a full-fledged criminal.

4-: From The Prahran Telegraph (Prahran, Melbourne, Victoria) of Saturday 15^th October 1904:

Francis Fuseo and William Fitzpatrick were charged with assaulting and robbing in company one William Phillips of 2s at South Yarra on 12^th October.
[…]
Francis Fuseo, jockey, 1 Wilson-street, South Yarra, on oath, said: I am innocent of the charge. At 20 to 12 last night I went to the coffee stall near the Federal Hall. Phillips was sitting down inside there. I walked in and hit him on the back and said “how is it Solly? Are you still at Jones’s?” He said “yes.” Others came into the stall and had a “bit of a pick at him” because he was drunk. With that he jumped up and said he would fight any one in the stall. Someone told him he could not fight his way out of a paper bag, and that made him worse.

5-: From the account of the cross-examination of Emma Edith Jolly in the suit Jolly v. Jolly heard in the Divorce Court, published in the Evening News (Sydney, New South Wales) of Wednesday 15^th March 1905:

Did he every [sic] carry out any of those threats you speak of, such as slashing a knife and getting the axe?—I got away from him. The week before I left he threw a block of wood at my head.
But it never struck you?—Yes, it did; it struck me in the broad of the back.
Did he throw it hard?—Yes.
Well, didn’t you have to go to the doctor?—No.
He didn’t hurt you, then?—Yes, he did.
Don’t you know he hasn’t strength enough to fight his way out of a paper bag?—All I do know is that it took three to hold him down.
His Honour: They don’t generally employ weak men as railway porters.

The phrase to fight one’s way out of a paper bag means to accomplish a very easy task in the following from Wheeney Philosophy, by ‘Tamerlane’, published in the Windsor and Richmond Gazette (Windsor, New South Wales) of Saturday 15^th June 1907:

“Crick seems to be playin’ the will-o-whisp, as usual. He’s a tough fellow to crack, that. They’ve had him in pretty well all the courts now, and he’s still goin’ strong, as we used to say when we was havin’ a good old ‘bust up’ round at old Bob Gee’s wine shop. I think Crick’s been afore every court in the land now—unless it’s the court martial; and it’s my private opinion he’d fight his way out of that court as easy as I’d fight my way out of a paper bag.”