This functionality has now been implemented in the base product--no more installing a separate add-on to provide it. (If you're using the add-on, it'll automatically be uninstalled when you upgrade to 5.1+)
The UX for the feature has changed somewhat from how the stash-archive add-on was laid out. "Download" no longer appears in the sidebar. Instead, it is in the actions dropdown next to the branch selector.
*Browse*

*Commits List*

*Branches*

It also appears in the actions dropdown for each branch listed

Last, but not least, when viewing any commit, you can now download the source at that commit.

The download functionality includes a full REST API which can be used to download any branch, commit or tag. It also supports filtering to specific paths and adding a prefix to paths inside the downloaded archive, as well as downloading in .tar or .tar.gz formats, as well as the default .zip. Documentation for the new REST endpoint will be published when 5.1 is released.
Best regards,
Bryan Turner
Atlassian Bitbucket
atlassian456, drorb,
Everything the archive plugin could do can also be done using the built-in functionality for 5.1+. The only changes are cosmetic, with "Download" no longer appearing in the sidebar (as I showed previously). Since it's a standard part of the application now, its REST functionality is documented with all of our other REST endpoints. This link should cover it.
Best regards,
Bryan Turner
Atlassian Bitbucket