Free solutions

ghettoVCB
ghettoVCB.sh - Free alternative for backing up VM's for ESX(i) 3.5 and 4.0+
ghettoVCBg2 - Free alternative for backing up VMs in ESX(i) 3.5 and 4.x (no SSH console required!)
This script performs backups of virtual machines residing on ESX(i) 3.5/4.0+ servers using methodology similar to VMware's VCB tool.
It's created and well maintained by lamw

Other script based solutions
Another ESXi backup script
http://communities.vmware.com/blogs/gabrielmaciel/2008/12/08/free-vm-backup-alternative-for-esx-35-and-esxi
Re: Free ESXi Backup Solution for Windows
Any new backup VM Solutions - ESX 3.5

VMware Converter Standalone
http://www.vmware.com/products/converter/
Is not designed as a backup solution, but could be used also for this purpose.

File copy solutions
For a powered off VM is possible to simple copy all the files in the VM folder.
To use also with a powered on VM you need to do create a VMware snapshot to unlock the *-flat.vmdk files
To make this copy several tools could be used:

• Datastore browser (directly from VIC or vSphere Client)
• SCP / Veeam fastSCP (need to enable SSH access: http://www.yellow-bricks.com/2008/08/10/howto-esxi-and-ssh/)

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VMware solutions (need license)

VMware Consolidated Backup (VCB)

VMware Data Recovery (VDR)
http://www.vmware.com/products/vcenter-data-recovery/

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Other commercial solutions

Veeam Backup & Replication
http://www.veeam.com/vmware-esx-backup.html
http://veeam.com/vmware-esx-server/backup/documents/Veeam_Backup_User_Guide_3.0.pdf
Veeam Backup & Replication is the first enterprise-ready solution that combines backup and replication in a single product for fast recovery of your VMware ESX servers. Backup is easy, but recovery can be hard – and that’s when the clock is ticking loudest. But with Veeam Backup & Replication, fast recovery is easier than you ever thought possible.

Vizioncore vRanger Pro 3.3
http://www.vizioncore.com/products/vRangerPro/vRanger3x.php
http://www.vizioncore.com/products/vRangerPro/documents/vRangerProUserManual_Eng.pdf
Vizioncre vRanger Pro 3.3 is the ideal solution for customers wanting the same reliability of vRanger Pro 4.0 DPP but require features that are not yet supported in the 4.0 version. Supports VMware 3.X, 4.0 (vSphere) and ESXi.

Trilead VM Explorer
http://www.trilead.com/
Trilead VM Explorer is a management tool that eases management, backup and disaster recovery tasks in your VMware ESX environment. Backups can be stored to ESX, Windows, Linux and FreeBSD based storage platforms or directly to a SAN. VM Explorer has only to be licenced once for your infrastructure, you don't have to pay any fees per ESX Server, CPU, or anything similar.

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Commercial solutions that may support ESXi in the future releases

esXpress
http://phdvirtual.com/products/esxpress-virtual-backup
esXpress radically alters the notion of how to protect data in virtual infrastructures in one simple way: we use the virtual infrastructure to backup and restore itself!
esXpress v3.6 is Compatible with VMware ESX v3, v3.5 and vSphere4.

Vizioncore vRanger Pro 4.0
http://www.vizioncore.com/products/vRangerPro/
http://www.vizioncore.com/products/vRangerPro/documents/vRangerPro-DPP-FAQ.pdf
vRanger Pro Data Protection Platform (DPP) is the market-leading backup, recovery and backup management solution for virtual environments. The 4.0 release is the next generation of vRanger Pro which features a completely redesigned architecture, including plug-n-play modularity and a service-based architecture. VMware ESX3.x and VMware ESX 4.x upon initial release. Future releases include support for: VMware ESXi 3.x, VMware ESX & ESXi 3.x VCB, VMware ESXi 4.x, VMware ESX & ESXi 4.x VCB.

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The free ESXi "problem"

With the free version of ESXi then you cannot actually use a third party tool to perform any backups. This is against your EULA. If it is the paid version of ESXi go for it.
So be sure which you are using to be sure you do not violate the EULA.

As for other mechanisms, you can snapshot the VM and then copy the data via the vSphere Client/VIC as well. There is also VMware Converter. These tools do not violate the EULA.

See also: http://www.virtualizationpractice.com/blog/?p=274 - VMware Forcing Third Parties to NOT Sell Products for Free ESXi

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Storage based solutions

If you have a stora NFS base, then file copy could be simple...
For other shared storage you can use storage snapshots.

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Backup ESXi configuration

You can use the RCLI to make a backup configuration of your ESXi host using
esxcfg-cfgbackup
You can find more information for the RCLI documentation here: http://www.vmware.com/pdf/vi3_35/esx_3/r35u2/vi3_35_25_u2_rcli.pdf

There are also other ways to do a backup of ESXi configuration, see:
http://www.r71.nl/index.php/kb/technical/212-vmware-esxi-configuration-backup
http://support.dell.com/support/edocs/software/eslvmwre/DG/ESXi_3_5_U4/HTML/app_c.htm