Time Machine is Apple's backup solution, included in OS X. An external drive is required. It backs up changed files every hour, and stores as many "snapshots" as it can hold. After the initial full backup, it only stores changes to files, which makes things more efficient, and allows for more granular snapshots. In addition to being able to do a full system restore, you can selectively restore older versions of specific files, which is handy in case you delete or make changes you didn't mean to.
Using Time Machine on OS X 10.7 and later also enables "local snapshots", which keeps snapshots of changed files on your main drive when you don't have your backup drive attached, provided there's sufficient space. Obviously this doesn't help if you have a drive failure, but in the case of accidental deletions, etc. it can be helpful.
For more information, Apple's Mac 101: Time Machine article is a good starting point.
Online Backup
If you're on the go a lot, you may want to consider an online backup service. BackBlaze is a popular one, but there's also CrashPlan, Mozy and several others.
The advantages are not needing an external drive, and having an off-site backup to recover from in the event of something nasty happening to your house. The downside is the monthly cost (I believe BackBlaze is a flat fee, others have differing schemes based on how much data you back up), as well as the time required to back up over the internet. These services only back up data that has changed, so after the initial full backup, things will get faster, but depending on your connection, it could be a bit problematic (particularly if you're travelling — most hotel Wi-Fi I've used isn't exactly speedy).
SuperDuper is a longstanding Mac favourite. It will make a complete clone of your drive to an external drive, which is then bootable and usable exactly like your internal drive. It also has a "smart update" feature, where after the initial clone, it only updates the backup with changed data, to minimize how long backups take. SuperDuper is the best way to get up and running again after a hard drive crash — you can boot off your backup drive right away and you'll your full system.
I'm listing this separately from the other online backup systems because it's not really a full backup, but it's definitely worth using. It's simply a folder that syncs to the Dropbox servers and wherever else you install your Dropbox client. Some limited storage of old versions of files is also included. 2GB is provided free, you can pay if you need more. The best use of Dropbox would be for your current working files. It won't store everything, but if you use it for the files you most often need, or those that you're using lately, you'll have up to date backups from whenever you have an internet connection.
Wrap Up
You'll have to do think about what exact setup suits your needs best. Ideally, a robust backup system would include at least two or three of these options in combination, so that you have multiple fallbacks. For example, you could do a daily or weekly SuperDuper backup, and then use Dropbox or Time Machine to restore the files you've been working on since your most recent backup. Adding in an online backup service for off-site redundancy will give you some added protection in the event your local backups are lost or destroyed.
Best Answer
I think the 'modern equivalent' of your anticipated disaster recovery scenario could still be done by CCC, with caveats.
Receive new machine, ready with Recovery partition & brand new OS, clone CCC boot partition back to new boot partition. Assuming parity with machine/drivers etc, i.e. like for like, then I foresee no issues.
The same maybe couldn't be said if the new machine was different, newer - as you cannot anticipate the new architecture - for that you'd need Time Machine, to be certain.