Can 1 gram hyaluronic acid really hold 6L of water?
Various figures can be found “on the internet”. E.g. ‘Hyaluronic Hydro Gel Has 1000 Times Water Holding Capacity Perfect for Hair to Heel’ (commercial link intentionally omitted).

Gel liquid phase absorption ability measure seems to be a ‘swelling capacity’ (ratio).

For hyaluronic acid (HA), it can be increased by additional crosslinking. In one paper from 1992, usual swelling ratio was found to be in the order of magnitude 1 to 10, the maximum figure was 120 (for crosslinking ratio much higher than 2.1; 25–30 for the ratio 2.1).[1]

In another paper, another guys are using biphasic nanoparticles (some composite of crosslinked and non-crosslinked HA), they state they have achieved for two different particle types ‘… swelling ratio 1270, 1000 %, respectively’[2] (note the percent sign). Which is, in both cases, approximately, not 1000× capacity, but rather 10× capacity, I'd say.

However, there are polymers with the claimed property – superabsorbents:[3]


  3.27 superabsorbent polymer
  
  Polymer that can absorb and retain extremely large amounts of a liquid relative to its own mass.
  
  Note 1: The liquid absorbed can be water or an organic liquid.
  Note 2: The swelling ratio of a superabsorbent polymer can reach the order of 1000:1.
  Note 3: Superabsorbent polymers for water are frequently polyelectrolytes.


References:


Shah, Chirag B., and Stanley M. Barnett. "Swelling behavior of hyaluronic acid gels." Journal of applied polymer science 45.2 (1992): 293-298.
Chun, Cheolbyong, et al. "Effect of molecular weight of hyaluronic acid (HA) on viscoelasticity and particle texturing feel of HA dermal biphasic fillers." Biomaterials research 20.1 (2016): 24.
Compendium of Polymer Terminology and Nomenclature (IUPAC Recommendations 2008)