How big was the wilderness in which John the Baptist lived and preached?

While the exact location of John's ministry is not known, we know that it was around the Jordan, with John 1:28 specifically placing it at "Bethany beyond the Jordan." The traditional location is places it just north of the northeastern corner of the Dead Sea. Jerusalem, the center of Second Temple Judaism, sits at the top of a verdant hill country. But as one makes the 10-mile downhill trip towards the Jordan, the land becomes a barren wasteland devoid of vegetation, except for the area immediately adjacent to the Jordan. This area extends from a few miles outside Jerusalem all the way through the Jordan Rift Valley until you hit the major settlements in modern day Jordan. The valley itself is about 10 miles wide (measurements taken from Google Maps).

Your question about how he managed to gain attention in a remote area is a good one. One thing to keep in mind is that the late Second Temple Period is a time of religious unrest, when many different groups and figures were competing for their vision of Judaism to be the dominant strain. Sects like the Pharisees and Sadducees, which focused on the study of Torah and Temple piety respectively, were generally centered in more populous areas (especially Jerusalem). However, there were more than a few other groups who made their home in the wilderness, including John, the Essenes/Qumran community, and eventually the revolutionaries of Masada. So John's location is not without precedent.

Bearing this in mind, along with Jerusalem's relative proximity to the area where John is supposed to have been preaching, we can paint a picture of a murmur among the people of Jerusalem about this man and his disciples in the nearby wilderness, enough of a murmur that the religious leaders came out to see what the commotion was all about. Three out of the four Gospels specifically mention people coming from Jerusalem (John 1:19, Mark 1:4, Matthew 3:5), which makes sense based on what we have said. What's perhaps more remarkable is not that the people of Jerusalem heard of John, but that Jesus did. From all the way in Nazareth, it's over 60 miles to get to that part of the country. Jesus enters this scene as the outsider, the country boy from the backwoods of the Galilee among a crowd of the educated and the urbanites.

The Judean Desert or the Wilderness of Judah is a small desert, approximately 1,500 square kilometers or some 600 Square miles and is very close to Jerusalem. Being so close to Jerusalem, it is no wonder that St. John the Baptist was able to have a large following.

The Judean Desert is bordered by the Mountains of Judea on the west and the Dead Sea on the East. It is considered a relatively small desert, spanning only 580 square miles, but it contains many fascinating nature reserves, historical sites, monasteries and primeval panoramas that make it an exciting and unique place to visit.

The Judean Desert is close to Jerusalem and relatively sparsely populated. The few settlements that are there were established at its perimeter. The desert is known for its rugged landscape, which has provided a refuge and hiding place for rebels and zealots throughout history, as well as solitude and isolation to monks and hermits. During the days of the Maccabees (about 2,000 years ago) large fortresses such as Masada and Horkenya were established in the desert. During the period of the great rebellion against Rome, the last battle of the Jewish zealots was fought on Massada, and during the period of the Second Temple, members of the Judean Desert cult lived there. - The Judean Desert

Here is where The Wilderness of Judea is located:

The edge of the wilderness is marked by 300 foot cliffs towering along the western shore of the Dead Sea and since its location is close to Jerusalem settlements can be found along its perimeters. Springtime in the wilderness is a time for blossoms, only for a very short season; for when the sun comes out in full force, it dries up again to become a dry wilderness. The Judean Desert wilderness in intertwined within scripture with some of the most significant events in Jesus’ life taken place in and around the Judean desert wilderness. It is located the southern province Judea in Israel; bordered by the Judean Mountains to the West and the Dead Sea to the East. Judea includes the territories of the tribes of Judah, Benjamin, Dan, Simeon, and part of Ephraim and contains the cities of Jerusalem, Bethlehem, Jericho, Bethany, Bethpage, Emmaus and others. Since Jerusalem is located on a hill anytime someone is traveling to Jerusalem they are “going up” and anytime they are going away from Jerusalem they are “going down”.

“The Judean Desert wilderness is nestled up against the Jordan river on its most north-eastern edge near where it enters the northern end of the Dead Sea. John the Baptist lived in this desert wilderness until the day of his manifestation to Israel: regarding in the days of Jesus when John the Baptist appeared preaching in the wilderness desert of Judea. Jerusalem and all Judea and all the country about the Jordan went out to hear him and were baptized in the Jordan by him, confessing their sins.” (Matthew 3:5-6).

Wilderness of Judea

Part of Judah E. of the Hill Country, descending to the Dead Sea. The wilderness of Judea extends from the waters of the Dead Sea to the very edge of the central plateau (or hill country), thus travelers from the E had to journey from five to eight hours through a waterless desert. Three well-watered spots are on its eastern edge, Jericho, 'Ain Feshka (10 miles S), and 'Ain Jidi (or Engedi 28 miles S). Three roads into Judea begin at Jericho; another road into Judea begins at 'Ain Feshka; and still another begins at Engedi. The roads from Jericho run NW to Ai and Bethel, SW to Jerusalem, and SSW to the lower Kidron and Bethlehem. Just after this last road crosses the Kidron it is joined by the road from 'Ain Feshka. The road from Engedi breaks into two branches, one running NW to Bethlehem and Jerusalem, a wild and difficult road never used by caravans, the other branch turning SW to Yuttah and Hebron.

Authorities say that the three features of Judea's geography that are most significant in her history are "her pastoral character, her neighborhood to the desert, her singular unsuitableness for the growth of a great city." Two, at least, of the prophets were born in the face of the wilderness of Judea-Amos at Tekoa, and Jeremiah at Anathoth. The wilderness was the scene of David's refuge from Saul; here John the Baptist prepared for his mission; and here our Lord suffered His temptation. - Israel in the First Century

Addendum:

In 2015, UNESCO’s World Heritage Committee added the archaeological complex at Al-Maghtas, Jordan—dubbed the Biblical “Bethany beyond the Jordan”—to its World Heritage List. The site has been venerated as the place where John the Baptist baptized Jesus since the late Roman–early Byzantine periods, when early Christians began making pilgrimages to the area. - Baptismal Site “Bethany Beyond the Jordan”