There are a little over a hundred and fifty books suggested here as a starting point. While it might seem like reading these books would take a lifetime, consider the numbers.
If you read one book from this list every month, you'll be done in around ten years. Read two books each month, and you'll be done in a little over five years. There's a good case to be made that investing five or ten years of your life into reading some of the greatest ideas in human history is a worthy undertaking.
Well, let's say you don't really buy that argument. I have good news for you: it'll likely take you far lesser time than you think to get through these books. You might go through the list, come across tomes like Gibbon's multi-volume Decline and Fall, and immediately be suspicious of such a claim.
However, consider the reverse. Most plays, dialogues, essays, and documents on the list can be read through in a few hours or less. Many multi-volume works such as Aquinas' Summa Theologiae need not be read in their entirety; a choice selection of chapters, or even secondary works summarizing important parts of the work can be read instead. Through the magic of the Internet, many plays and books can be absorbed through easily-accessible movies and recorded performance, or in the form of audiobooks.
The real challenge with reading the great books all by yourself instead of a classroom environment is that you miss out on the peer pressure that forces you to read, and the classroom discussions where you're forced to think by having to articulate your ideas.
While it's hard to transfer peer pressure over the internet, we can address the second aspect to some extent. Where possible, I've included links to supplemental commentaries that provide additional insight and the relevant background to understand the book. These are not essential reads, but are available if you want to take a deeper dive into a particular book.
Just take things one baby step at a time. This is totally doable, and you can do it.