https://www.dndbeyond.com/sources/dnd/dmg-2024/bastions#SpecialFacilityDescriptions

I know the special facilities exist, but as written you don’t buy them, they’re awarded at certain levels.

True but they do seem to require space. Upgrading space costs money.

Quote from Wysperra >>

True but they do seem to require space. Upgrading space costs money.

 
The space for the facilities is acquired as part of the mechanism; there’s a couple that you can pay to have upsized for more features, but I’m not sure there’s any of those in the first selection.

A Cramped space is free and what you start with. To get any of the level 5 (The OP is level 6) facilities, it needs to be upgraded to Roomy (500 gp). Some of the roomy space have upkeep or investment costs. The garden is one of those you mentioned which if you can upgrade from Roomy to Vast which will cost 2000 gp.

Is it worth it? Depends. But it is a way to spend a lot of $$ and comes with benefits.

Quote from Wysperra >>

A Cramped space is free and what you start with. To get any of the level 5 (The OP is level 6) facilities, it needs to be upgraded to Roomy (500 gp). Some of the roomy space have upkeep or investment costs. The garden is one of those you mentioned which if you can upgrade from Roomy to Vast which will cost 2000 gp.

Is it worth it? Depends. But it is a way to spend a lot of $$ and comes with benefits.

 

No, you don't need to spend gold to acquire space for special facilities. If you refer to the section you linked, it specifically says that special facilities are not purchased, they're simply gained at the appropriate level. 

Unlike basic facilities, special facilities can’t be bought; a character gains them through level advancement. At level 9, a character gains two additional special facilities of their choice for which they qualify; they gain one additional facility at level 13 and another at level 17. The Special Facility Acquisition table shows the total number of special facilities in a character’s Bastion. Each new special facility immediately becomes part of the character’s Bastion when the character reaches the level.

Clearly you need to find aurora's whole realms catalogue and order 3,000 GP in cheese. 

Interesting. It makes no sense. It means they included an upgrade cost from Cramped to Roomy for no reason. Nothing fits in a cramped space.

Quote from Wysperra >>

Interesting. It makes no sense. It means they included an upgrade cost from Cramped to Roomy for no reason. Nothing fits in a cramped space.

 Basic Facilities includes upgrade costs that serve no mechanical benefit - purely for aesthetics and RP. 

Special Facilities that are able to be upgraded do get mechanical benefits. For example, upgrading the Special Facility: Garden will take it from the baseline of Roomy and increase it to Vast, which will double the harvest that it produces every 7 days.

Of the level 5 options, only the Barrak, Garden, and Workshop can be upgraded, which all start at Roomy. None of the Special Facilities that can be upgraded in size at any time start at Cramped. Only the Special Facilities that cannot be upgraded and the Basic Facilities include Cramped spaces.