This much to the foolish argument of the folks roistering around with the "there is no free place anymore" argument. If TUI is able to purchase 11 km2 inmidst of the highly populated Italy, Gay State would be able to purchase an equally large area somewhere in less densely populated place. Taking a density of 20.000 persons per km2, those 11 km2 can give home to 220.000 persons. Considering that in Brazil (and elsewhere) more extended areas could be purchased without evicting many inhabitants, and that population density in modern city districts easily can reach some 40.000 persons per km2, what doubts can someone have that a territorial Gay State can be realistically established somewhere? As one can see, the price for the area was rather acceptable.
A tourist trap is more acceptable to the public and political mind than a Gay Nation. We may be able to purchase the land, even setup an initial base - but once the public is aware of what is going on there will be a huge outcry.
And yes there is a lot of land out there for the taking, in snow country to where you can only raise crops part of the year. In desserts where water resources are limited, on islands where water and land is limited, in many areas. The good arable land with basic resources is owned and claimed.
Yes the population density of a city can be pretty high. Take San Francisco - in a 25 square mile area (5 miles by 5 miles) you have 739,426 people (living - more who work there every day).
The problem is not the ability to put a lot of people on a small plot of land - the problem is that you still need a lot of land in order to grow food, provide industry, provide power, provide sewage and provide trash removal, fresh water and all of that.
San Francisco imports water from the Hetch Hetchy reservoir which is a glacial valley in Yosemite National Park in California. The water is piped/canaled 150 plus miles (241 KM) to S.F.
Food is imported not only from the central valley of California (50 to 100 miles/80 to 160 km), but also globally. Each person requires 600 square meters (6458.346 square feet) of land to raise food, supply roads to the city, for electrical, sewage and landfill (That is the minimal, many people actually have more land for their use)
Using San Francisco as example: although it sits on 5 miles x 5 miles (25 square miles/65 square kilometers), it uses 171.3 square miles (443.66 square km) (13 miles by 13 miles roughly on top of the 5x5) to supply all of those people. That is assuming the bare minimum of land use per person. The reality is more like twice to three times the amount of land since we are talking about a people who want a lot more than what they need.
Increase the density of the urban setting and you need more supporting land.
Unless you desire to have the GLBT nation at the mercy of supporting nations (for imports of basics like food and water) then you need to rethink buying land and claiming it as a Nation.
We can cut some of those numbers down if we rethink how crops are grown, if we place the amount of proteins in the diet off of cattle/meat and into other sources. The best I can get from land is 250 square feet (23.2 square meters) per person for
food only (small animals, organic intensive year round crop production of agricultural (land based) produce substantially backed by other food resources (fish). I am basing these numbers on my ability to raise a year supply of vegetables on a small plot of garden (18' x 16') for two people along with my estimated land use for each person for chicken, grains (wheat, rice, etc) and some (like half of the usual diet) of beef. Even then we are talking of a spartan diet - one that most people would not like or even desire. Filling and good for you, but not heavy in meats and dairy.
My actual garden does not supply fruits (oranges, peaches, berries) with the exception of one grape vine which produces a lot of grapes and in reality has less growing land since I use raised beds with a center path for tending the garden. I use organic, intensive, successive planting - 3 season growing methods (living in warm sunny California we have a much longer growing period) I raise mostly seasonal vegetables. I can/preserve some of the crop, but usually we live on fresh straight out of the garden food.
This is the extreme method of living off of the minimal amount of land - it would not work on a large population - not unless you want everybody working a garden to raise most of their own foods.
And it does not include supply roads from the farms, wells or fresh water supplies, sewage treatment, land fill/recycling facilities, energy production etc. Nor any of the luxuries like plastics, metals, etc.