The English language has long and short vowel sounds. Short and long vowels also exist in some other languages. And it's interesting to know when native English speakers come to let it be the Czech Republic or Hungary (countries which have short and long vowels in their languages), can they define, when they hear the speech of the natives, whether it's a short or long sound?
English historically did have "long" and "short" vowels—meaning vowels distinguished only by the length of time they were pronounced for. The ancestor of modern "tub" had a short vowel
/u/ (sometimes written ŭ), while the ancestor of modern "tube" had a long vowel
/u:/ (sometimes written ū).
However, somewhere roughly around Shakespeare's time, English went through a change called the "Great Vowel Shift", which completely mixed up the long vowels while leaving short vowels mostly intact. Most long vowels turned into diphthongs (approximately "sequences of two vowels"), and now that something other than length was distinguishing them, the length difference itself disappeared.
So nowadays, an English "long A" is actually a diphthong
/ej/, a "long I" is a diphthong
/aj/, and a "long U" is a diphthong
/ju/. Historically, the distinction was length—but that hasn't been true for a few centuries now.
(A few dialects do still have phonemic length distinctions in certain contexts, such as Australian English Mary~merry, mentioned by curiousdannii. But these are edge cases that have nothing to do with "long vowels" as described in English classes.)
Thus, the answer to your question is no. What English-speakers call "long vowels" have nothing to do with actual phonetically long vowels, as used in Hungarian, Japanese, and so on.
can they define, when they hear the speech of the natives, whether it's a short or long sound?
Maybe, maybe not: individuals vary. Although English does not have tone, English speakers can learn Chinese, and some of them can hear and produce tone differences very early in their exposure to Chinese. The underlying question seems to be whether knowing vowel length in one language enables you to distinguish vowel length in another language.
This was studied in Tsukada 2011, comparing perception of Arabic and Japanese vowel length by speakers of Arabic, Japanese and Persian (Persian does not have a length contrast). It turns out that Japanese and Arabic speakers perform asymmetrically. Although in both cases speakers are better at identifying vowel length in their own language (not surprising), Japanese speakers are better at discriminating Japanese vowel length than Arabic speakers are at discriminating Arabic vowel length. Moreover, Arabic speakers are no better than Persian speakers at identifying Japanese vowel length, and Japanese speakers are not any better at identifying Arabic vowel length than Persian speakers are. However, this depends on the kind of test you use to determine "can they hear?".
Most English accents don't actually have long vowels, except for some like Australian English where "Mary" and "merry" differ only in vowel length.
What are commonly called long vowels in English are actually diphthongs, which are vowels which start as one vowel and gradually glide to a different glide. For example, the vowel in "high" starts with a low "a" sound and changes to a higher "i" sound. We'd transcribe "high" as [hai] showing the starting and ending sounds of the diphthong.
In general most people will be able to hear most diphthongs in other languages even if they don't recognise all the individual vowel sounds because most diphthongs are not between similar vowels, but ones that cross the mouth: high to low or front to back etc.