If you accept the Genealogical Proof Standard of a reasonably exhaustive search, you are still left with the question of what is reasonable and where do I search? The Standard uses the phrase, "Assumes examination of a wide range of high quality sources." This is done so that, quoting again, the search, "Minimizes the probability that undiscovered evidence will overturn a too-hasty conclusion."
Now, why did I say "if you accept the Genealogical Proof Standard?" Simple. First, you have to know the Standard exists. Very, very few of the researchers I help each week are even vaguely familiar with the existence of a Genealogical Proof Standard. The reason is relatively simple -- the novice researcher has little or no contact with the genealogical community and even if they are more acquainted with the community, they could go to conferences and seminars for years and never hear the Standard mentioned. One of the obvious reasons is that the Board for Certification of Genealogists does not advertise itself much in the media and is seldom has official representatives at any of the local conferences. For example, I just attended the BYU Family History and Genealogy Conference and although there were six Certified Genealogists teaching classes, I heard no mention of the Genealogical Proof Standard in any class I attended.
Personally, I am not aware of any better statement of the goals of genealogical research than the Genealogical Proof Standard and I think it would immensely benefit the genealogical community if the Standard were more widely known. Now, back to the questions in the title of this post.
What is reasonable? The field of law has grappled with this concept for hundreds of years. The consensus definition is what a reasonable person would do in the same or similar circumstances. So the question becomes is it unreasonable to ignore records and sources that may prove probative of facts concerning an ancestor? The answer is yes. It is unreasonable to ignore reasonably available records and sources. By the way, everything on the Internet could be considered reasonably available given the time and expense that finding those online records would have taken in the not too distant past.
For example, from a question posed by a commentator, what kinds of records would she search to find a birth date? Think of it this way, during your lifetime, how many forms have you filled out that required your birth date? The last time I picked up a prescription at the pharmacy, the clerk automatically asked me my birth date to verify my identity. Now project that into the past. How many records do you think your ancestor left that might have contained his or her birth date? More than one? Maybe hundreds? Here is a very selective partial list of documents that contain a birth date or year:
Birth records, including birth certificates when available and church christening records, confirmation records, baptism records, marriage records, divorce records, school records, military records of all kinds and types, tax records, the list goes on and on and on.
So how many of these records constitutes a reasonably exhaustive search? I submit, all of them.
Where are these records found? Some of them may actually be online. But the majority of them are likely sitting somewhere near where the person was born or died. So guess what? You might just have to plan a research trip and go look for some of the records, but only after you have exhausted your online possibilities. Where can you go for suggestions of the types of records that might be available? A good place to start is the FamilySearch Research Wiki for any given area of the world.