How to repeat:

• Shut down your Internet connection.
• Launch Steam. A window will appear, notifying about connection error. It has three buttons, "Retry", "Go to Offline Mode" and "Exit".
• Press "Exit". Steam will not shut down, it will start in Offline Mode.
• Exit Steam. Enable Internet connection.
• Launch Steam again. Steam will act like there's no Internet, and show "Connection Error" window again.

My specs:

Processor Information:
    Vendor:  AuthenticAMD
    Speed: 2300 Mhz
    2 logical processors
    2 physical processors
    HyperThreading:  Unsupported
    FCMOV:  Supported
    SSE2:  Supported
    SSE3:  Supported
    SSSE3:  Supported
    SSE4a:  Supported
    SSE41:  Unsupported
    SSE42:  Unsupported

Network Information:
    Network Speed:  

Operating System Version:
    Ubuntu 12.04.1 LTS (64 bit)
    Kernel Name:  Linux
    Kernel Version:  3.2.0-35-generic
    X Server vendor:  The X.Org Foundation
    X Server release:  11103000

Video Card:
    Driver:  ATI Technologies Inc. AMD Radeon HD 6570M/5700 Series

    Driver Version:  4.2.11931 Compatibility Profile Context
    Desktop Color Depth: 24 bits per pixel
    Monitor Refresh Rate: 59 Hz
    VendorID:  0x1002
    DeviceID:  0x68c0
    Number of Monitors:  1
    Number of Logical Video Cards:  1
    Primary Display Resolution:  1366 x 768
    Desktop Resolution: 1366 x 768
    Primary Display Size: 13,54" x 7,60"  (15,51" diag)
                                            34,4cm x 19,3cm  (39,4cm diag)
    Primary VRAM Not Detected

Sound card:
    Audio device: 

Memory:
    RAM:  2945 Mb

Miscellaneous:
    UI Language:  English
    LANG:  ru_UA.UTF-8
    Microphone:  Not set
    Total Hard Disk Space Available:  297340 Mb
    Largest Free Hard Disk Block:  247618 Mb

Installed software:

Recent Failure Reports:
    Thu Dec 27 11:27:00 2012 GMT: file ''/tmp/dumps/assert_20121227132700_1.dmp'', upload no: ''Couldn't resolve host name''