It's SS 3.1.13, Translatable 2.0.8 and TDO dev-master.
I have two DO classes: "Facility" ($has_one) and "FacilityType" ($has_many) with which I chose to be managed in ModelAdmin (as they are global properties of the site owner). I cannot however get TranslatableDataObject extension to enable translation for them, despite reading carefully the setup instructions: https://github.com/bummzack/translatable-dataobject

Here's my config:
// Set the site locale
i18n::set_locale('en_US');

SiteConfig::add_extension('Translatable');

// Set allowed locales
Translatable::set_allowed_locales(array('en_US', 'pl_PL'));
TranslatableDataObject::set_locales(array('en_US', 'pl_PL'));
Facility::add_extension("Translatable");
Facility::add_extension("TranslatableDataObject"); // tried "TranslatableDataObject('Title')" as well
FacilityCategory::add_extension("Translatable");
FacilityCategory::add_extension("TranslatableDataObject");

Could anyone please confirm that this setup should or won't work for me?

Ensure to have Translable module installed too, and to set_default_locale, not just set_locale.

https://github.com/silverstripe/silverstripe-translatable/blob/master/docs/en/index.md

I'm not too experienced with these modules, but it looks like it should work. The best way is to try it and see :)

I'm so sorry - no go. It did not help.

Hmm, it should work but just in case: try setting the extensions via the YML config

Also, have you added translation content for the locale that isn't the default? You need to add this via the CMS first before it will load - otherwise it will probably just default to the default locale.