The Space Elevator monument will be available to build once you've unlocked 6 unique buildings. Each unique building has requirements you must meet before they unlock. The buildings and requirements are as follows:
Statue of Wealth - City money: $500,000
Transport Tower - Public transport has 1,000 weekly passengers
Grand Mall - Construct an airport
Opera House - City extracts a total of 3,000 units of natural resources
Expo Center - 2,000 tourists visit the city
Stadium - City produces a total of 1,000 units of goods
Once those buildings have been placed in your city, the Space Elevator will become available. It costs $1,500,000 to build and $16,000 a week for upkeep. It will generate a massive influx of tourists into your city.