All content on the internet is stored on computers not unlike the one you have at home, but with a lot more hard disk space and usually more processing power as well. These are your web servers. Companies maintain huge rooms full of servers and sell access to that storage to other companies. Since engineers are coming up with more efficient storage all the time, you can have a huge amount of data storage in a very small physical space. It is ridiculously cheap to store data nowadays. So no, the chance of running out of space is relatively low. A bigger concern is the durability of the data. Hard drives can degrade over time and are susceptible to data loss from physical disturbances (say, an earthquake or simply dropping it). That means a good data center is built with redundancy in mind, maintaining multiple copies of the data and even copying it to analog storage medium like tape.
Edit: Software also keeps getting better at storing more information with less “data units”, i.e. bytes thanks to clever compression algorithms. So the space you need to store a high-resolution image today is a lot less than what you would have needed 10 years ago.
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