Exchange & Cloud Email

I have ~10 email accounts and ~30 GB PST files which are synced into my local (desktop) outlook, am using X1 (x1.com) to search in my emails.
I would like to make all data accessible (including search) on my other devices (e.g. laptop, tablet, iphone) without manually configuring it in each device.

What is the best way to do it? Which cloud service is most suitable?

You need an IMAP service like e.g. FastMail. Their webmail is very fast, including search and recently it is also available as an app for Android and IOS. No configuration required so. If you need more than 30 GB then a Premier account suits you best.