Professions
This feels very plausible at first because 'Flying' points to 'Pilot' and 'Public speaking' points to 'Motivational Speaker'. However, the theme breaks on 'Heights' (as it's not a profession, but a characteristic), 'Spiders' (an arachnologist is a job but isn't the immediate connection), and 'Open spaces', which has no clear corresponding profession.
Extreme sports or activities
An idea sparked by 'Flying' (skydiving) and 'Heights' (rock climbing). This theory falls apart because 'Spiders', while a fear for some, is not an activity, and 'Public speaking' is the opposite of an extreme sport, being an indoor, stationary task.
Things associated with anxiety
This is close, but too broad. 'Public speaking' and 'Flying' are classic anxiety triggers, but the clues point to a more specific type of established, named fear. The connector is more precise than general 'anxiety', which could include things like deadlines or social gatherings not covered by the clues.