It depends on the field.
My ongoing PhD work is in applied mathematics, where English is pretty much all you need. Sure, knowing some Russian, French and German can be useful, if you want to read various older seminal papers in their original form, but the actual results from those papers have usually been widely reproduced (often in a more modern and readable form) in textbooks written in English. For more modern research in my field, English is pretty much the universal choice of language, perhaps the only halfway significant contender being Chinese.
(In pure mathematics, the situation is somewhat different, partly due to the "longer half-life" of pure math papers noted by Anonymous Mathematician in their answer. Indeed, typically the non-English papers one occasionally encounters in applied math tend to be on the theoretical side.)
I'm also an amateur cryptographer, and within that field the dominance of English in academic writing is even more complete. This is partly due to historical reasons: the modern academic study of cryptography, as opposed to clandestine military research and occasional hobbyist dabbling, is a relatively new field, and emerged from applied mathematics and computer science at a time when especially the latter field was strongly dominated by the U.S. and other English-speaking countries. Basically, within crypto, the only thing you may need languages other than English for is historical (meta)research.
On the other hand, in the last few years I've also dabbled in the humanities, specifically in assyriology. Within that field, at least rudimentary knowledge of French and German is all but essential (some of the central reference works and dictionaries are written in those languages), and Latin isn't completely useless either. And of course, you also need to learn the ancient languages that you're studying (for me, that's Akkadian, some Sumerian and a bit of Hittite so far), and familiarity with their neighbors and (ancient or modern) relatives can be pretty useful, too.
So, yes, that's at least one field where knowing only English would be a significant impediment. I'm not saying it would completely prevent you from studying the field, but it does deprive you of access to some fairly useful sources and references. And besides, the general attitude in assyriology seems to be that a researcher should be polyglot as a matter of course. For example, the Reallexikon der Assyriologie (an important encyclopedia of the field) happily mixes together entries written in German, English and French, depending on each contributor's preferred language, and expects the reader to be able to make sense of any of them.