For me the defining characteristic of HD800 are:
1. Super wide soundstage.
2. Precise imaging
3. Fast decay ( no ringing).
4. Smooth frequency response.
5 Good bass distortion and excellent mid & highs distortion.
6 Huge earcups.
Smooth doesnt meant no peaks, the HD800 has peaks but they are smooth and easily equalized. Also another defining characteristic is clean CSD waterfall, that is fast decay. The decay speed is directly linked to FR smoothness so its mathematically redudant to list it independently yet I do becose not many people know this. Imaging is also result of smooth frequency response, therefore points 2 and 3 are really just different effects of point 4.
Soundstage too is result of frequency response, theoretically you could give any headphone any soundstage width with right EQ but I never got soundstage as wide as HD800 with EQ. I managed to widen my HD58X but it wasnt "super wide".
HD800S is overhyped, it has far inferior distortion to HD800 and the high frequency peak can be easily modded or EQed. The HD800 has one of the best dynamic driver distortion levels, out of full sized DD open back headphones, I only saw it get beaten by Focal Utopia and its cheaper derivates and 50mm biocelluIose driver headphones like Audioquest Nighthawk but those are more of semi open.
I love the huge earcups, its like cavern for ears, there is nothing else like it. HD800 is extremly unique headphone, there is no cheaper alternative, however I do have some cheaper similiar-ish suggestions.
1. HD560S, pretty wide soundstage, smooth FR fast decay. Its kind of like HD800 but not as low distortion and not as wide sounstage, however at 240 g weight and 150€ price, its great choice.
2. AKG K701, a budget headphone with super wide soundstage. Distortion is worse than HD58X & HD560S, about as good as HD600 in bass, it has spike in midrange distortion. Frequency response sucks balls and is pretty rough. Its dirt cheap and comfortable. K7XX, 712 PRO and K702 65th Anniversary have superior distortion but different pads which to me completly ruin soundstage making it about as wide as cheap closed on ears.
3. HD700, I tried them sadly after I got sudden permanent hearing loss so I dont know how they sound. From reading alot, they seem like little less wide HD800 with HD650 like distortion and very unpopular ear piercing harsh FR. I say, go with HD560S, better distortion, lighter and better comfort, better out of box FR and far cheaper. Maybe HD700 is little wider but for that price I would get used HD800.
The Philips Fidelio series is said to have wide soundstage, the distortion is very bad, I have never tried them. They are inexpensive and people seem to like them alot.