How to transfer a Minecraft world from one computer to another?

I have a world on Minecraft that I've had for the past two and a half years. I've made a lot of progress on it, and it has been on survival mode the entire time it has existed.

However, the computer that it is on is five years old, and I feel it's near it's final days. I really don't want to lose my Minecraft world. There's too much work and nostalgia I have from it to let a dead computer take it away from me.

How can I transfer this world to another computer?

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