A 503 error is a "service unavailable" error, meaning the server is unable to serve up the page you're looking for, usually due to heavy traffic. Many websites mitigate this issue by using CDNs, or content delivery networks, which have redundant copies of the same content across multiple servers, so that if one server becomes too busy, the request can be handed off to another. This keeps high traffic websites afloat during peak times.
Reddit, however, cannot benefit from this strategy as much as, say, Facebook can. Reddit is almost entirely database driven with very dynamic content. All the traffic to Reddit hits the database, as it needs to load comments, titles and post text. "High bandwidth" static media items like images and videos are hosted on imgur, gyfcat or youtube, which can distribute that data via CDN. But you can't put user comments on a CDN because they change so frequently, there would be a time delay if you waited for all the redundancies to propagate. And if you're updating the redundant often enough to be close to real time, you're hitting the database pretty hard, so this is likely not happening either.
You might wonder how Facebook can avoid this issue, since they, too, deal with a lot of database driven content like comments and text posts. While their exact data model is not exactly publicly available, there's a few plausible solutions they're using. Most likely, Facebook uses a relational database model where different tables are stored across different servers. Furthermore, when you browse Facebook, Facebook determines which content you see in your feed, allowing their servers to pull content from parts of the database that aren't getting hit as hard. Furthermore, while Facebook has a massive user base, each user only views a relatively small chunk of Facebook data at a time. Facebook has 1.23 billion active users each month, but the typical user is only loading content from fewer than 1000 friends, this allows Facebook to strategically move data around to different servers, alleviating the load on a central database. Even on popular posts on Facebook, you're only given the most recent comments, unlike Reddit where the entire comment history of a thread is loaded at once. Reddit aggregates a specific piece of content to a much, much larger audience than a single post on Facebook gets. Reddit has a front page, where tens of thousands of users are actively viewing, commenting and voting.
TL;DR: Reddit's data must be in "real time" and due to the large number of users hitting the site at once, we often see 503's because the server can't handle the load. Larger sites like Facebook mitigate this issue by separating data onto different servers, which is feasible since a particular piece of content on Facebook goes to a generally much, much smaller audience than a particular piece of content on Reddit.
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