Impeachment of Obama is possible, and there are some legitimate grounds on which he could be impeached. Regardless of what someone's 'opinions' are, the grounds for impeachment are not only enumerated in the Constitution, they are also fairly open ended and pretty much up to what the House of Representatives may decide at any given time. Just for an example, 'failure to uphold the Constitution' which is a sworn oath, is wide open to interpretation. Practically any president could be impeached by an opposing House of Representatives.
Now, the practical matter is that the House is highly reluctant to bring about impeachment charges against anyone and especially so the sitting President. The reasons are many, but chief of those would be the tit for tat retaliation that would likely ensue far into the future. Such possibility would render the office of President to a practically useless position as all the executive would ever do is fight the charges. Then there is the matter of the Senate, the current Senate is controlled by the President's party, and they are the ones that would actually try the case. (House brings the charges, Senate conducts the trial) Ergo, even if impeached, as Clinton was, there is no way the case would be resolved in removing Obama from office by the Senate. So there is no real taste to start a fight that in the end no one could benefit from.
Now, impeachment is not reserved for the President, it is possible to impeach and try anyone in the government, like certain IRS agents and managers... real substantive changes could be realized in going that route!