I would like to know if Jacobian adjustment is required for the following situation.
my model has parameters whose “typical” values are either very small and very large. Following through Stans manual about prior choice, https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations, what I have been doing is to convert both large and small parameters on a unit scale. For example, if the typical value of beta is 3.8x10^-10, and the typical value of eta is 20000, then these are changed into

beta_raw = log(beta/3.8x10^-10) and
eta_raw = log(eta/20000)

No because the priors are appropriately put on beta_raw and eta_raw. But you should do a bunch of simulations to see if the implied prior distribution of beta and eta have reasonable dispersion and skewness.

Another clarification. Suppose instead of the above transformation for eta, I use lognormal distribution for eta, do I include jacobian adjustment. Specifically, the model is now coded as

With the lognormal distribution for eta, posterior mean for eta is close to its true value. However, I wasn’t sure if jacobian adjustment is required for this case or not.

If you priors pertain to things in the parameters block then no Jacobian adjustment is needed. Conversely, if your priors pertain to something not in the parameters block, then a Jacobian adjustment is needed unless it is constant in which case you don’t need it for inference but would need it for things like Bayes Factors.