You have asked a much debated question that has been discussed since the first showing of this excellent play. It is interesting that Chekov himself referred to this play as a "farce," whereas many other critics argue that the element of tragedy in this play far outweigh the comical elements. Certainly, the characters in the play are shown to be unable to act to help themselves and are unable to embrace or accept the immense social changes experienced by Russia in the time of the play. Although the play doesn't fit the traditional definition of a "tragedy," there is definitely a tragic element to it, and the way in which the characters are depicted and their utter helplessness leads some to argue that this is not a farce.
On the other hand, there are undoubtedly farcical elements to the play. Characters such as Leonid Gayev, and his constat reference to imaginary billiard shots, and Simon Yepikhodov, with his many calamities and the love triangle of which he is a part, undoubtedly bring a comic element to the play. And yet, interestingly, in spite of these comic elements, it is still possible to regard each of the characters and their situations as having elements of both comedy and tragedy. Thus it is clear that it seems problematic to classify this play overall as either a tragedy or a comedy. There are definite elements of both, and Chekov seems to have delibeately robbed us of the ability to easily classify his work into a discrete category. Perhaps this play represents his belief that it is possible to find humour even in bleak and hopeless situations.