There is a website, Earliest Known Uses of Some of the Words of Mathematics. Some entries are,
DIOPHANTINE ANALYSIS (named for Diophantus of Alexandria) occurs in French in a letter of March 1770 from Euler to Lagrange: “ce problème me paraissait d'une nature singulière et surpassait même les règles connues de l'analyse de Diophante” (“this problem appeared to me to be of a singular nature and surpassed the known rules of Diophantine analysis”).
Lagrange used “analyse de Diophante” in a letter to D’Alembert in June 1771.
“Diophantine Analysis” occurs in English in the chapter title “Demonstration of a Theorem in the Diophantine Analysis. By Mr. P. Barlow, of the Royal Military Academy, Woolwich.” in The Mathematical Repository, New Series, Volume III (1809) page 70.
[This entry was contributed by James A. Landau.]
DIOPHANTINE EQUATION. Felix Klein used Diophantische Gleichungen in “Die Eindeutigen automorphen Formen vom Geschlechte Null” in the 1892 issue of Nachrichten (page 286): “Die Relationen kann man in Diophantische Gleichungen umsetzen, welche dann leicht übersehen lassen, unter welchen Umständen Multiplicatorsysteme möglich sind, und in welcher Anzahl.” [James A. Landau]
Diophantine equation appears in English in 1893 in Eliakim Hastings Moore (1862-1932), "A Doubly-Infinite System of Simple Groups," Bulletin of the New York Mathematical Society, vol. III, pp. 73-78, October 13, 1893 [Julio González Cabillón].
Henry B. Fine writes in The Number System of Algebra (1902):
The designation "Diophantine equations," commonly applied to indeterminate equations of the first degree when investigated for integral solutions, is a striking misnomer. Diophantus nowhere considers such equations, and, on the other hand, allows fractional solutions of indeterminate equations of the second degree.
DIOPHANTINE PROBLEM. The phrase "Diophantus Problemes" appears in 1670 [James A. Landau].
The OED2 has the citation in 1700: Gregory, Collect. (Oxf. Hist. Soc.) I. 321: "The resolution of the indetermined arithmetical or Diophantine problems."
I interpret the question as not being about the problems of size in category theory in general and how to deal with them (which are fairly well-understood and the subject of other questions on this site), but about what Eilenberg and MacLane actually meant in their original paper. The phrasing of that particular footnote is sloppy, but I think section 6 of their paper ("Foundations") suggests that what they meant is that "this category would lead to paradoxes if we required the objects of a category to form a set rather than something like a proper class".
My guess is that they used the word "aggregate" in the definition in section 2 as a nod to the fact that to be formal, one may want to take these to be proper classes (or something related), but assumed that the average mathematician reading the paper would interpret "aggregate" as "set" at least until they got to section 6. So they added a footnote pointing out that they were aware of the issue, but deferred a fuller discussion of it (and an explanation of what "aggregate" can formally be defined to mean, or other ways one can deal with the problem while still interpreting "aggregate" as "set") to the later section. For instance, in section 6 they wrote "we have chosen to adopt the intuitive standpoint, leaving the reader free to insert whatever type of logical foundation (or absence thereof) he may prefer".
Best Answer
Norman Steenrod is generally credited with coining the phrase. I'd call it "tongue-in-cheek" rather than sarcastic. See http://en.wikipedia.org/wiki/Abstract_nonsense and references there.