"Go and take a running jump"

Eric Partridge, A Dictionary of Slang and Unconventional English, second edition (1938) has the following entry for the phrase:

go and take a running jump at yourself! (Rare in other moods.) Go to blazes!: a c[atch-]p[hrase] (C 20) expressive of scorn. E.g., in John G. Brandon, The One-Minute Murder, 1934. Cf. go and play trains! [which Partridge defines as meaning "Don't bother me!"]

Partridge also has entries for "go and boil your head" and "go and eat coke," so the "go and" part of the phrase seems to have been a recurring formula for introducing a dismissive recommendation.

The phrase "take a running jump" simply refers to attempting an informal long jump (as opposed to a standing broad jump, which involves leaping from a standstill). However, an interesting example from Canada suggests that "taking a running jump" might be used to imply "running headlong [into something] without exercising appropriate caution" or, more simply, "acting precipitously." From Official Report of Debates, House of Commons [of Canada] (1932) [combined snippets]:

Our complaint—and we will continue to make it is that our friends on the other side seem willing to take a running jump at almost any policy and put it into effect, and then sometimes take a step backward and make some corrections, or at other times—and more frequently—to stick to it through thick and thin, with resulting detriment to the people affected.

A Google Books search turns up several relevant matches for "go and take a running jump" from the 1950s. From an unidentified article in Medical Press (1950) [combined snippets]:

Obviously a harassed doctor with his burden of work increased will find it harder to devote time to the personal side of his relations with his patients. How much of a compensation is it to him to know that if he tells Mrs. Sofa-Saint to go and take a running jump at herself he is jeopardising only 15s, a year instead of £15?

A Mr. Dillon of the Irish Parliament uses the expression on multiple occasions, in 1950 the in 1952. From Parliamentary Debates [of Ireland]; Official Report (1950) [combined snippets]:

Mr. Dillon: I said in this House that I regarded a proposal of that nature from the Northern Ireland spinners as impudent and insolent and that I told them to go and take a running jump at themselves.

...

I shall tell them to go and take a running jump at themselves.

And from Parliamentary Debates [of Ireland]; Official Report (1952) [combined snippets]:

Mr. Hickey: It is the system you should condemn.

Mr. Dillon: I understood their difficulty and we talked it over. I sent them back to that fellow with a message to go and take a running jump at himself and if he jumped too far he might land where he never intended to land. He did not jump.

An earlier, related instance appears in an unidentified article in Blackwood's Magazine (1936) [combined snippets]:

P.S.—Jim Watt wants you, as a personal favour, to put your scheme in your pocket and take a running jump into the river. There's a place Peter Croft calls The Dump-Hole where you can make a dandy splash.—A.P.M.

A search of the British Newspaper Archive yields two matches for "go take a running jump at himself" from 1931 and 1932, both of which appeared in the same London newspaper. From "Mr. (or Miss) Wamble," in the [London] Bystander (June 17, 1931):

The Last of Their Line--No. 5 Any guy here that ain't all out to make it snappy from the word Go can just go take a running jump at himself.

And from "Cedric Belfrage discusses the Passing Picture Shows," in the [London] Bystander (September 7, 1932):

Mrs. B. also rises to tell us that whenever her husband returns to civilisation he is always hugely appreciative not only of her companionship, but of the most trivial creature-comforts, which the ordinary husband would take for granted. So old Schopenhauer can just go and take a running jump at himself.

Conclusion

The earliest print instances I found of the invitation to "go and take a running jump at [oneself]" are from 1931 and 1932 in a London newspaper. Partridge identifies the expression as a catchphrase less than a decade later, and I see no reason to doubt him on this point.

One feature of catchphrases (most memorably itemized by Charles Mackay in Memoirs of Extraordinary Popular Delusions and the Madness of Crowds [1852]) is that they need not possess any historical point or clarity of meaning in order to capture the popular imagination—hence, Mackay's observations about the successive vogues of "Quoz!" and "There he goes with his eye out!" and "Has your mother sold her mangle?" during the first half of the nineteenth century.

Perhaps the phrase "go and take a running jump at yourself" originated as an extemporaneous expression of scorn that caught on specifically because it doesn't really make sense.