Is it possible to have a picklist in data extensions in Marketing Cloud, for example a field called City then users can filter using a drop down and select from London, Manchester or Birmingham?
Unfortunately this isn’t supported as natively as you might want. Even on objects synchronised from Sales Cloud, fields that are pick lists in SC are only provided as text fields in MC.
You will need to do some custom configuration, where you create a separate DE, holding all the options for your pick list fields, and reference it in e.g. Cloud Pages, using LookupOrderedRows to populate drop down input field options.
If this is the use case you are in need of supporting, I can elaborate more on the solution if needed. If you need the pick list values to be available within MC, e.g. when working with Data Extensions in Contact Builder - I’m afraid there is not much we can do there.
Bit of an update since 2018... picklists weren't available for a long time in Marketing Cloud and unfortunately still are not out-of-the-box (OOTB).
However, meanwhile, it is possible through certain third-party solutions (like this one) to define picklists in an extra admin panel and then use these picklist values when filtering your data. will probably help to make sure data entries/filtering happens more accurately. A bit of explanation on how that works can be found here. Essentially it's an extra "semantic" layer which will still use text as a field format underlying it in SFMC.
Since many customers now sync SF CRM (service cloud or sales cloud) to SFMC, that helps in making sure values are used consistently (for reporting, preparing campaign data,...)
For an overview of (OOTB) SFMC supported field formats (known as "data types"), look here.