I guess if the OP really intended the ‘Beethoven and Brahms’ question, then the answer is yes, such pastiches are being written every hour of every day, as routine mundane exercies by music students the world over. And a few who passionately believe that this music is the ‘right’ way to do things. But the latter never produce something that can compare with anything more than the mundane everyday products of the great composers. Creating a new masterpiece in an old style is their holy grail, and is appropriately elusive.
CLARIFICATION: The use of the word “classical” refers to music written by such musical luminaries as Beethoven, Mozart, Bach, and so forth. Stuff with titles like “Symphony in E major”, or “Concerto No. 5”, etc. No lyrics. Requires lots of people wearing tuxes and other clothing finery to play. Needs to hire a large hall or other cavernous locale to present. Uses lots of instruments that don’t require electricity. The kind of music that causes students in a high school music “apreciation” class to moan and groan.
There. Let’s try this again.
Yes: see Adams, Corigliano, Shulamit Ran, etc. (The Chicago Symphony Orchestra always has a Composer in Residence; ditto many other big-city orchestras.)
There are also composers who straddle the Gray Divide between “classical” and “pop”: Glass, film scorers, and dilettantes like Paul McCartney and Billy Joel. So you can get pretty gray.
But if you mean, serious artists who work in the medium of music rather than paint, and survive on grants and commissions, and compose “pieces” for orchestras and quartets and opera companies, etc., then yes, there are still plenty of those around. The top of that heap is occupied, nowadays, by (IMHO) Corigliano and Adams.
Yeah, okay, Beethoven’s Symphony #5 in C Minor is pretty hackneyed these days. But you can still bring the house down with an old chestnut like the Verdi Requiem or Tchaikovsky’s 1812 overture*.
*) Provided you use cannons.
YES!
I presume none of those (other than the last 
) are meant to be completely comprehensive.
But yes, there’s plenty of people writing such pieces, for such performances. As an example, pieces going by such titles in the Proms in London this summer are:
Henze - Symphony no. 10
Corigliano - Clarinet Concerto
Casken - Symphony (Broken Consort)
(And you can listen to them online 
)
What about Anne-Sophie Mutter? I was in Tower Records one day and saw a poster of her and . . . sort of lost track of time. Later, I was finally able to tear my eyes away.
Anne Sophie Mutter may be the better violinist, but she lacks the Swedish-Pornstar looks of Leila Josefowicz
This is a good question, and one that may warrant a thread of its own. It’s been addressed here somewhat, but the discussion seems to have generated more heat than light. It’s tough to define, to nail down exactly, but the same is true for lots of genres. There are works that quite obviously are classical, and those that clearly are not; but it’s tricky to figure out exactly where the border is. If somebody attempts to take one of the classical forms and write a work in that form (e.g. a symphony, a sonata, a concerto, a fugue), and they know what they’re doing and how that form works, I figure that’s enough to call it “classical.” But not all “classical” music fits into some previously recognized form.
And it’s not a matter of quality. We do use the word “classic” to describe something that has stood the test of time and is recognized for high quality, but that’s obviously not the sense in which the OP meant the term. “Classical music” does indeed refer to a specific style or genre, and “new classical music” or “bad classical music” are not necessarily contradictions in terms.
Classical music is a tradition, as I think someone has already pointed out. For any work of art(/entertainment/culture/imagination) and its creator (e.g. composer), it’s possible to talk about their influences, who they imitated or learned from or evolved from or reacted against or are similar to. For something to be considered “classical” music, you ought to be able to trace its influences in a fairly direct chain to works that are unquestionably “classical” (Bach, Haydn, Mozart, Beethoven, etc.).
One characteristic of classical music as opposed to other kinds (though not a defining characteristic) is that it is planned out and written down. Though it needs performers to bring it to life, the fundamental artistic merit is in the composition, not in the performance. In this respect it’s closer to a Shakespeare play than it is to a pop song.
There’s also a sense of progression, of the music going somewhere. To fully appreciate the piece, you have to start listening at the beginning and follow through to the end. This isn’t necessarily the case with, say, a film score or a jazz piece or New Age or poppy instrumental music.