The are on Earth several such large areas enclosed by mountains. The key is that such large basins are not enclosed by one mountain range, but by two or more.
The largest such geographical feature is the endorheic Tarim Basin. It is enclosed by the Tian Shan mountains to the north, the Pamirs to the west and the Kunlun range to the south; most of the basin is a deadly desert, the Taklamakan. Today, the Tarim basin is in the People's Republic of China.
But enough of the Tarim; with all its glorious history and eerie beauty, it is too far away, and too dry, and much too exotic.
In this answer I will describe the Pannonian Basin, also known as the Carpathian Basin; but you can also look at a physical map of Asia Minor, for another example in the same general area.
Geographically, the basin consists of the Pannonian Plain, divided into a smaller north-western part and a larger south-eastern part by the low Transdanubian Mountains. The basin is surrounded by mountains, specifically the Western Carpathians to the north, the Inner Eastern Carpathians to the east, the Southern Carpathians to the south-east, the Dinarides to the south-west, and the Alps to the west. Sometimes, mostly for political reasons, but not utterly implausible geographically, the Transylvanian Plateau is included in the Basin; in this case, the Inner Eastern Carpathians can be seen as a mountainous island and the eastern boundary of the Basin is formed
by the Outer Eastern Carpathians.
The Basin, most of which is in Hungary with small parts in Slovakia, Romania, Serbia and Austria, is about 450 km west to east and 300 km north to south; if some of the relevant portions of the surrounding mountains are included its extend grows to about 500-550 km west to east and 350-400 km north to south. With the Inner Eastern Carpathians and the Transylvanian Plateau included, the maximum extension of the basin would be about 750 km (470 miles) from Vienna to Brașov (also known as Kronstadt).
[Physical map of the Pannonian Basin, CC BY-SA 3.0, map created by Koba-chan from DEMIS Mapserver, which are public domain.]
The Danube enters the Basin from the north-west through the Devín Gate and exits to the south-east through the Iron Gates. If the Transylvanian Plateau is included, then there is another river which pierces the moutainous ring surrounding the basin: the Olt, which crosses the Southern Carpathians through the Red Tower Pass.
The arc of the Carpathians, which forms the northern and eastern boundary of the Basin, is part of the Alpine orogeny; it was raised at the beginning of the Cenozoic (formerly known as the Neozoic), about 60 million years ago, when the African Plate and the Eurasian Plate collided; the Carpathians were raised as some microplates pushed against others: full details are complicated, but see for example "Geodynamics of Carpathian and Crimean fold belts formation" by Y. Krupsky and Y. Koltun.
[Schema of the Alpine Orogeny. By Woudloper - Own work, CC BY-SA 1.0.]
The Basin was originally part of the Paratethys Sea; it eventually became isolated from it about 10 million years ago, and the Pannonian Sea disappeared in the Pliocene, about 4 million years ago, leaving behind a flat plain dotted by island mountains.