In the 90s it was inconvenient to be out and about if you were expecting a call. You might be able to check your messages though, by dialing your answering machine from a payphone.
Payphones barely exist anymore, so it would be even more inconvenient. If you need to be reachable at all (nowadays by email/sms), a cell phone is nearly a requirement. If you do not need to be reachable, you still put yourself at a significant disadvantage by not carrying one.
What is your life like where you would not be inconvenienced by not having a cell phone?
Donald Knuth gave up email in 1990. Too distracting. (I dunno but I wouldn't think he has a cellphone)
When I started grad school in '07 I wanted to forgo a cellphone. I couldn't. My advisor was even annoyed I held out on a smartphone for as long as I did (caved in 2010).
In 2013, sick of my smartphone I went back to a dumbphone. It was great until 2015 when I started working at a place so crappy I have to routinely tether my phone for productive internet access.
My point: it's very possible to live w/out a cell phone. Frankly it's not even hard (I "forget" my phone at home). However, unless you're tenured or are otherwise your own boss, you'll be forced to have one.
Mostly pleasant and stress-free. I haven't owned any phone in ~6 years.
What is your life like to where you think not having a cell is impractical.
In any case a smart phone is fundamentally a weak point in privacy/security no matter which one you get.