More entertainment in this thread than Netflix with the recurring characters and themes
Every fan will have their opinion but this is my personal ranking of achivements:
1) Premier League - The bread and butter for any top club and the most accurate gauge of a team's tactical, footballing, physical and mental strength as it's 38 games over 9 months against all opponents in all situations. Brian Clough once said he'd happily accept a League Title in exchange for first round exits in the European Cup, FA Cup and League Cup, and I'm kinda aligned to that view.
2) Champions League - Competing against the continent's best over a fairly large sample size in front of massive worldwide audiences.
3) CL qualification and a domestic cup
4) CL qualification
5) Domestic cup - sad to say but they just don't have the same allure due to expansion of European competitions.
The well resourced PL clubs, Big Six, or whatever name you wish to give them, are primarily judged on the first two priorities by fans and the media. The question is over how long a period of time ? 5 years in fairness is more than most modern managers at top clubs are given - look at how Roman Abrahmovic mercilessly dumped managers at Chelsea even after winning big prizes (which isn't my preferred model).
Arsenal do remind me of how we used to be under Gerard Houllier and Rafa Benitez. We lacked the guile to unlock low blocks especially at home in the league, and so never won the title. We constantly dropped points to teams like Charlton, Portsmouth, Birmingham etc. However we were superbly organised and our style of play lent itself well to European competitions so I wouldn't be surprised if Arsenal won the CL. They were very streetwise the other night and coped well in the febrile atmosphere of Madrid.
However making that next step in the League will depend on the quality of the attacking reinforcements in the summer.