It's pretty easy to do. You can see how big it is in relation to Saturn's rings during the arrival cinematic for the new Crucible map, The Dungeons.
Saturn's rings are 73,000 km wide. From the cinematic, the Dreadnaught is about 0.07 times as long, and in the same plane as the rings. That makes the Dreadnaught...
[b]5110km long![/b]
That's over 5 million metres.
From other, better quality images of the Dreadnaught, I've calculated that one side of one of its square ends is about 0.3 times as long as it's length. Therefore, that gives a total volume of...
[b]1.2 x 10^10 (12000000000) cubic kilometres![/b]
So, yeah. It's pretty damn big.
[b]Edit:[/b]
Obviously, I am not expecting that we will be able to visit even a tiny fraction of this. Just like how we cannot visit all of the Earth, Venus, etc.
Also, I find it likely that Bungie does not actually intend for it to be this big, and instead either didn't realise (or expected other people not to realise) just how big Saturn's rings are. That, or it's just to make it look more badass.
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Another eerie factoid to add is that with the Hive's ability to manipulate dimesional space the actual interior of the Dreadnaught could be much MUCH bigger, i mean look at Crotas pocket dimension on the moon, we drop down into a darkened area through a portal that leads to castle floating in nothingness with its own dead star but from other perspectives the "area" we are falling into is just a pit that ends near Phogoths summoning chamber. Food for thought while exploring lol.